Shortest paths and geodesics - DiVA portal609061/... · 2013-03-04 · duction to the general...

Shortest paths and geodesics

in metric spaces

Nicklas Persson

Umeå UniversitetHandledare: Linus Carlsson

Examensarbete: 30 högskolepoängHT 2011

SHORTEST PATHS AND GEODESICS

IN METRIC SPACES

Nicklas Persson

Nicklas Persson: Shortest paths and geodesics in metric spaces,Master’s thesis in mathematics, c©November 2012

Institution:Mathematics and mathematical statistics

Supervisor:Linus Carlsson

Examiner:Per-Anders Boo

Location:Umeå, Sweden

Time frame:November 2012

Abstract

This thesis is divided into three part, the rst part concerns metric spaces

and specically length spaces where the existence of shortest path between

points is the main focus. In the second part, an example of a length space,

the Riemannian geometry will be given. Here both a classical approach

to Riemannian geometry will be given together with specic results when

considered as a metric space. In the third part, the Finsler geometry will

be examined both with a classical approach and trying to deal with it as

a metric space.

Sammanfattning

Denna uppsats är indelade i tre delar. Den första behandlar metriska

rum med betoning på längdrum där existensen av kortaste vägar är hu-

vudsyftet. Den andra delen ger ett exempel på ett längdrum, Riemanska

geometrin. Här kommer både en klassisk upplägg till den Riemanska geo-

metrin att ges tillsamanns med resultat där den är betraktad som ett

metriskt rum. I den tredje delen betraktas Finslergeometrin utifrån både

i ett klassik upplägg och ett försök ges att behandla den som ett metriskt

rum.

vi

Acknowledgements

I would like to thank my supervisor Linus Carlsson for his many hours

spent helping and listening to my many questions and my examiner Per-

Anders Boo who has taken time reading through this thesis and pointing

out errors and commenting on how to improve it. Without their help the

result would not have been nearly as good. Also I wish to thank Alexander

Zdunek for his time spent reading this work in its early state and coming

with constructive criticism.

vii

Contents

1 Introduction 1

1.1 Basic notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Metric geometry 3

2.1 Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Length spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Shortest paths and geodesics . . . . . . . . . . . . . . . . . . . . 13

3 Riemannian geometry 25

3.1 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 Riemannian metrics . . . . . . . . . . . . . . . . . . . . . . . . . 443.3 Riemannian connections . . . . . . . . . . . . . . . . . . . . . . . 473.4 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4 Finsler geometry 59

4.1 Finsler manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.2 Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.3 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Appendices 73

A Metric spaces 73

B Topological spaces and topology 79

B.1 Topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 79B.2 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

C Algebra 105

C.1 Linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

D Vector spaces 113

D.1 Dierential forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

E Functions 133

F Euclidean geometry 137

G Notation 139

H Bibliography 143

viii

List of Figures

1 Example of a non-length space . . . . . . . . . . . . . . . . . . . 72 Example of another non-length space . . . . . . . . . . . . . . . . 83 A sequence fn for n = 1, 2, . . . , 6 . . . . . . . . . . . . . . . . . . 104 A geodesic γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 An atlas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 A unit sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 A tangent bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 A vector eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 Domain of germs . . . . . . . . . . . . . . . . . . . . . . . . . . . 3910 Tangent bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6211 A continuous function . . . . . . . . . . . . . . . . . . . . . . . . 8212 A counting of the rationals . . . . . . . . . . . . . . . . . . . . . 9313 An element of a base . . . . . . . . . . . . . . . . . . . . . . . . . 9514 A non convex set . . . . . . . . . . . . . . . . . . . . . . . . . . . 10315 Domain and codomain of a function f . . . . . . . . . . . . . . . 133

ix

1 INTRODUCTION

1 Introduction

In this thesis, an introduction to the theory of metric geometry will be given.Metric geometry deals with the geometrical topics which has a relation to thenotion of distance. During the 20:th century a lot of the theory developed ingeometry concerned the analysis of manifolds but in later years a lot of the workdone in the dierential geometry has shown that a metrical approach can beconducted to reach the earlier results.

The results in this thesis are divided in three chapters. The rst gives an intro-duction to the general theory of metric geometry with emphasis on length spaces,the existence of shortest paths and geodesics. The following two chapters gives abrief classical approach to Riemannian geometry and Finsler geometry togetherwith attempts at trying to deal with them as metric spaces and studying theexistence of shortest paths. Some material which does not directly relate tothe main results of this thesis are collected into seven appendices which can befound directly after the three main chapters where the last Appendix G gives alist of the notations used here. The thesis end with an index and a bibliography.

An experienced reader of the topics covered can skip the material in the ap-pendices or fast skim trough the material but readers with less experience canwith benet start by reading the appendices recommended before each chapterand whenever an unknown term is used, look it up where it can be found in theappendices.

1.1 Basic notation

Here will be given some basic notation used in the thesis. Vectors will be denotedby [1, . . . , n] and not to confuse vectors of the form [a, b] with closed intervalsbetween a and b, the non-conventional denotation I[a,b] will be used for suchintervals. The letter d will be used for dierent metric functions, γ correspondsto dierent curves and l(γ) is the length of the curve γ.

An open ball will be denoted by Bd(x, r) where d corresponds to the metric, xthe center of the ball and r is the radius. Closed balls will in a similar mannerbe denoted by Bd(x, r). By Rn, the set of n-tuples of the real numbers R isintended and En corresponds to the Euclidean space which is Rn together withan Euclidean structure which is basically an Euclidean inner product, norm andmetric.

The letter M will denote a manifold, TpM will denote the tangent space of Mat a point p and TM the tangent bundle of M . The function g will be used forRiemannian metrics and the corresponding object in the Finsler geometry, theFinsler structure is denoted by F . For notations not covered in this short guide,the list of notations in Appendix G is recommended.

1

2 METRIC GEOMETRY

2 Metric geometry

In this chapter there will be given a short introduction to the theory of metricspaces with emphasis on the theory of length spaces and geodesics. The goalof this chapter is to prove the Hopf-Rinow-Cohn-Vossen theorem which givesa couple of equivalent statements for which a length space has shortest pathsbetween every pair of points.

To read and understand this chapter some prerequisite knowledge of topology,topological spaces and metric spaces are needed. For those who are not so fa-miliar with these terms or need refreshing these topics Appendix A deals withmost of the knowledge needed for metric spaces and Appendix B contains thetheory for topological spaces and topology. The goal of the appendices in thisthesis is to make it understandable for a reader who has studied two semestersof mathematics without having to use exterior material. For those interestedin more detailed studies in the subjects of this chapter and the aforementionedappendices the book [BBI01] deals with metric geometry, where most of theresults in this chapter can be found and the books [Rud76], [Mor05] are usefulfor metric spaces and topology. Where [Mor05] is a good starting point for rsttime students of these subjects and [Rud76] for the more advanced.

In this chapter the standard topology (see Denition B.1.6) will be assumed fortopological spaces.

2.1 Paths

To later dene length spaces and the notion of geodesics the concept of paths isessential. Paths are continuous maps of intervals and important to observe notthe image of the map. Formally they are dened as following.

Denition 2.1.1 (Path). A path in a topological space X is a continuous mapf : I[a,b] → X.

The almost equivalent term to path, curve (see Denition B.2.7) which is moregeneral will also be used for paths in this chapter.

Example 2.1.1 (Path). Given the topological space R2. The curveγ : I[0,1] → R2, given by:

γ(t) =

x = t2, t ∈ I[0,1]

y = t3, t ∈ I[0,1]

(2.1.1)

is a continuous map between the points [0, 0] and [1, 1]. So γ is a path between[0, 0] and [1, 1].

Example 2.1.2 (Path). Given the topological space R2. The curveγ : I[0,1] → R2, given by:

3

2.1 Paths 2 METRIC GEOMETRY

γ(t) =

x = t2, t ∈ I[0, 13 ]

x = t, t ∈ I( 13 ,

23 )

x = 2t+ 23 , t ∈ I[ 23 ,1]

y = t3, t ∈ I[0,1]

(2.1.2)

is not a continuous map because the map is discontinuous at t = 13 and therefore

γ is not a path.

The concept of equivalent paths is important to be able to speak of convergingpaths later on. Heuristically two path should be equivalent if the collection ofpoints visited by the paths are the same and the points are visited in the sameorder. Furthermore if one of the paths stops at one point for a time while theother continues but otherwise are similar then they should be considered equal.Combining these properties and formalizing them gives the below denition.

Denition 2.1.2 (Equivalent paths). Two paths γ1 and γ2 are said to be equiv-alent if they belong to the same equivalence class dened by the equivalencerelation:

γ1 : I[a,b] → X and γ2 : I[c,d] → X are equivalent whenever there exists anondecreasing and continuous map θ : I[a,b] → I[c,d] such that:

γ1 = γ2 θ (2.1.3)

Remark 2.1.1. Paths of the same equivalence class are called parametrizationsor re-parametrizations of one another and each are of the same length.

Example 2.1.3 (Equivalent paths). The paths γ1 : I[0,1] → R2, where

γ1(t) =

x = t

y = t(2.1.4)

and γ2 : I[0,2] → R2, where

γ1(t) =

x = t

2

y = t2

(2.1.5)

are equivalent.

Let θ : I[0,1] → θ : I[0,2] be dened by: θ(t) = 2t. This continuous map satises:γ1 = γ2 θ

Example 2.1.4 (Equivalent paths). The path γ1 : I[π4 ,−3π4 ] → R2, where

γ1(t) =

x = r cos(t)

y = r sin(t)(2.1.6)

4

2.2 Length spaces 2 METRIC GEOMETRY

is a parametrization of the quarter-circle

x2 + y2 = r2. (2.1.7)

Another equivalent path of this quarter-circle is γ2 : I[−1,1] → R2, where:

γ2(t) =

x = tr√

1+t2

y = r√1+t2

(2.1.8)

That this is another parametrization can be seen by implicit derivation on

x2 + y2 = r2 which gives:ddx (x2 + y2 = r2) = 2x+ 2y dydx = 0⇐⇒dydx = −xy

Setting −t = dydx gives that: x = yt, y = x

t , inserting these in x2 +y2 = r2, solve

for x and y gives (2.1.8).

2.2 Length spaces

The intrinsic metric or length metric is a metric possible to dene on everymetric space. For this metric the distance between two points is the length ofthe "shortest path" between these these points. The term shortest path will bedened later and is in fact crucial for the understanding of geodesics.

Denition 2.2.1 (Intrinsic metric). Given a topological space X with a metric.The intrinsic metric (length metric) dI(x, y) for x, y ∈ X is given by:

dI(x, y) = infγ∈P

l(γ) (2.2.1)

Here P is the set of paths from x to y and if there is no path in X of nitelength between x and y, then set dI(x, y) =∞ and l(γ) is the length of γ.

Denition 2.2.2 (Shortest path). Given a curve γ : I[a,b] → X. If γ is a pathand l(γ) ≥ l(γ) for every path γ ∈ X such that the end points of γ are γ(a) andγ(b). Then γ is a shortest path.

Proposition 2.2.1. The shortest path between two points x, y ∈ En is given bya straight line between x and y.

Proof. Assume there exists a shorter path γ between x, y than the straight lineγ. From the denition of the length of a curve B.2.8 we get the following equa-tion:

l(γ) = supa=t1<t2...<tn=b

n−1∑i=1

d(γ(ti), γ(ti+1)) (2.2.2)

5


But the partition t1 = a, t2 = b is the straight line between x, y and sincel(γ) is the supremum of the possible partitions: l(γ) ≥ d(x, y). This gives acontradiction and therefore γ is a shortest path.

Example 2.2.1 (Intrinsic metric). Given the topological space E2. In E2, theshortest paths are straight lines due to Proposition 2.2.1. The intrinsic metricdI between the two points x = [0, 0] and y = [3, 3] is given by dI(x, y) =

√18.

When the intrinsic metric is dened, the concept of length spaces is straightfor-ward when the intrinsic and "usual" metric on the given space coincide. Furtheron in this thesis almost every space considered will be length spaces and whendealing with the question if shortest paths exist between dierent points in aspace, the whole theory in this chapter build on it being a length space.

Denition 2.2.3 (Length space). Given a topological space X with a metricd. The space X is a length space if for every x, y ∈ X:

d(x, y) = dI(x, y), (2.2.3)

where dI(x, y) is the intrinsic metric.

Example 2.2.2 (Length space). The Euclidean space En is a length space. Asseen in Proposition 2.2.1 the shortest path in dI(x, y) is a straight line betweenx and y and in this case the Euclidean metric d(x, y) coincides with dI(x, y).

Example 2.2.3 (Length space). The sphere S1 ⊂ R2 with the Euclidean metricis not a length space. This can be seen by observing that paths in S1 are partsof the sphere while the Euclidean metric gives rise to "chordals" between pointswhich are not on the sphere and d(x, y) ≤ dI(x, y) if x 6= y(see Figure 1).

Example 2.2.4 (Length space). Given a connected subset X in E2. If X isconvex the set is clearly a length space but if X is not convex than it is not alength space. Then there exist two points x, y ∈ X such at there do not exists astraight line between them which is in X. For these points dI(x, y) 6= d(x, y) asseen for example in Figure 2.

From these examples the reader should now have a feeling of what length spacesare and that the distance between points from the metric in length spaces caninformally be thought of as being measured inside the space.

A curve being natural is a property dependent of the parametrization of thecurve. Natural curves will be used in this chapter as a theoretical tool amongother things to prove the Arzela-Ascoli Theorem 2.3.1. What makes these typeof curves interesting beside simplifying certain computations are that for a givencurve there exists a parametrization which is natural as is shown in Proposition2.2.2.

6


Figure 1: Example of a non-length space

Denition 2.2.4 (Natural curve). A curve γ : I[a,b] → X is natural (unitspeed) if:

l(γ(c), γ(d)

)= d − c for every c, d ∈ I[a,b], where by l

(γ(c), γ(d)

)means the

length of the curve segment from γ(c) to γ(d).

Remark 2.2.1. The name unit speed parametrization for a natural parametriza-tion comes from the fact that:

d

dtl(γ(a), γ(t)) = 1. (2.2.4)

A parametrization which satises:

(γ(a), γ(t)) = c(t− a) (2.2.5)

is called a constant speed parametrization of speed c.

Example 2.2.5 (Natural curve). The curve γ : I[0,1] → R2, where

γ(t) =

x = t

y = tis not a natural curve. For example

7


Figure 2: Example of another non-length space

ddt (γ(0), γ(t)) =

= ddt

√(t− 0)2 + (t− 0)2 =

√2 6= 1, for t > 0.

But this parametrization is instead a constant speed parametrization with speed√2.

The notion of uniform convergence is a cornerstone in analysis and will be usedrepeatedly later on. The dierence from pointwise convergence is informallythat converging uniformly has to do with how it converges over all of its domainand for pointwise it is sucient that it converges at every point.

Formally pointwise convergence is stated as: Given a metric space (X, d) anda subset Y . The sequence of functions fn : Y → X is said to be be pointwiseconvergent on Y to the function f : Y → X if for every x ∈ Y :

there exists, for every ε > 0 an N such that for every n ≥ N ,

d(fn(x), f(x)) < ε. (2.2.6)

This will further on be denoted as:

limn→∞

fn(x) = f(x). (2.2.7)

8


Denition 2.2.5 (Uniform convergence). Given a metric space (X, d) and aset Y . A sequence of functions fi, where fi : Y → X is uniformly convergentwith limit f : Y → X if:

For every ε > 0, there exists an N ∈ N such that for every x ∈ Y and n ≥ N ,

d(fn(x), f(x)) < ε. (2.2.8)

Example 2.2.6 (Uniform convergence). Given the set R with the Euclideanmetric. The sequence of functions fn,

fn : I[0,1] → I[0,1] (2.2.9)

where fn = xn is not uniformly convergent. This function instead satisfy theweaker condition of point-wise convergence and converges to

f(x) =

0, if 0 ≤ x < 1

1, if x = 1. (2.2.10)

Assume that fn converges uniformly, choose ε = 12 . Then there should exist

an N ∈ N, such that:

|fn(x)− f(x)| < 1

2(2.2.11)

for every x ∈ I[0,1] and all n ≥ N .

But choosing:

1 > x >1

2

1N+1

and n = N + 1. (2.2.12)

gives a contradiction due to:

|fN+1(x)− f(x)| = |xN+1| > 1

2(2.2.13)

so fn is not uniformly convergent.

Informally it can be seen that fn does not converge uniformly by inspecting thegraphs (see Figure 3) of fn and noticing that a problem will arise due to thediscontinuity at x = 1 which will cause that for every ε < 1 there is an N suchthat for x "close" to 1:

d(fn(x), f(x)) > ε (2.2.14)

The uniform convergence will now be used to dene uniform convergence ofcurves which is crucial to prove the Arzela-Ascoli Theorem 2.3.1 later on.

9


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 ≤ x ≤ 1

f n (x)

Plot of fn (x)

n=1n=2n=3n=4n=5n=6

Figure 3: A sequence fn for n = 1, 2, . . . , 6

Denition 2.2.6 (Uniform convergence of curves). A sequence of curves γiconverges uniformly to a curve γ if γi has a parameterization with domainI[a,b] which uniformly converges to some parametrization of γ with domain I[a,b].

Example 2.2.7 (Uniform convergence of curves). Given the curveγ : I[0,1] → R2, where

γ(t) =

x = t2

y = t. (2.2.15)

The sequence of curves γn : I[0,1] → R2, where

γn(t) =

x = t2 + 1

n cos(t)

y = t(2.2.16)

converges uniformly to γ.

Example 2.2.8 (Uniform convergence of curves). Given the curveγ : I[0,1] → R2, where

γ(t) =

x = t2

y = t. (2.2.17)

The sequence of curves γn : I[0,2] → R2, where

γn(t) =

x = t

2

2+ 1

n2

y = t2

(2.2.18)

converges uniformly to γ.

10


In this example the curves γn converges uniformly to a limit, which is a re-parametrization of γ and this is sucient for γn to be uniformly convergent.

The aim of the following theory is to achieve results for when shortest pathsand the closely related term geodesics exists between dierent points in lengthspaces. The two following propositions state the already mentioned result thatevery curve has a re-parametrization which is natural and with a small modi-cation of the proof that the curve has a re-parametrization with arbitrary speed.

Proposition 2.2.2. [BBI01] Given a rectiable (l(γ) <∞) curveγ : I[a,b] → X. It can be represented by:

γ = γ1 γ2, (2.2.19)

where γ1 : I[0,l(γ)] → X is a natural curve and γ2 : I[a,b] → I[0,l(γ)] is nonde-creasing and continuous map.

Proof. For constructing γ1, let γ1(τ) be the point on γ such that the length ofthe interval from its origin and to γ1(τ) is τ .

Now dene γ2 as:γ2(t) = l

(γ(a), γ(t)

)(2.2.20)

for all t ∈ I[a,b]. This map is nondecreasing, continuous and its set of values isI[0,l(γ)].

The curve γ1 can be constructed by choosing a t ∈ I[a,b] for every τ ∈ I[0,l(γ)]

such that γ2(t) = τ . Then dene γ1(τ) = γ(t) and thus γ1 : I[0,l(γ)] → X.

To show that γ1 is continuous, let τ1 = γ2(t1) and τ2 = γ2(t2). Now γ1(τ1) andγ1(τ2) are end points of a path γ′ which is the curve segment from γ1(τ1) toγ1(τ2) of γ. The length of γ′ is:

l(γ′) = l(γ(t1), γ(t2)) = γ2(τ2)− γ2(τ1) = τ2 − τ1. (2.2.21)

Utilizing that the distance between the endpoints in a length space is shorterthen the length of the path from Proposition 2.2.4 gives:

d(γ1(τ1), γ1(τ2)) ≤ |τ1 − τ2| (2.2.22)

From (2.2.22) the map γ1 is Lipschitz continuous, this implies continuity byTheorem A.0.9 and the below Remark A.0.9 and since γ′ is a re-parametrizationof the path γ′1 which is the curve segment from γ1(τ1) to γ1(τ2), the length ofγ′1 is:

l(γ′1) = l(γ(t1), γ(t2)) = τ2 − τ1. (2.2.23)

This concludes that γ1 is a natural re-parametrization of γ.

11


Proposition 2.2.3. Given a rectiable curve γ : I[a,b] → X. It can be repre-sented by

γ = γ1 γ2, (2.2.24)

where γ1 : I[0,c·l(γ)] → X is a constant speed curve of speed c > 0 and γ2 :I[a,b] → I[0,l(γ)] is nondecreasing and continuous map.

Proof. The proof is similar as the proof of Proposition 2.2.2. Choose

γ2(t) = l(γ(a), γ(t)) (2.2.25)

as in Proposition 2.2.2 which will then be a nondecreasing and continuous mapand γ1(τ) = γ(t), where τ = cγ2(t).

The following proposition gives two important properties for paths in lengthspaces. These properties will be used repeatedly in dierent proofs in this chap-ter.

Proposition 2.2.4. [BBI01] Given a length space (X, d) and a pathγ : I[a,b] → X. Then the following statements are true:

a) l(γ) ≥ d(γ(a), γ(b)) (Triangle inequality)

b) Given a sequence of rectiable paths γi which converges pointwise to γ.Then lim

i→∞inf l(γi) ≥ l(γ) (Semi-continuity of length)

Remark 2.2.2. The operator lim inf called the limit inferior is dened for asequence xk as:

limk→∞

inf xk = supk∈N

infi≥k

xi. (2.2.26)

This can be thought of as the smallest limit of a subsequence or if no such exists±∞.

Proof.

The a) part of the proof is essential the same proof as for Proposition 2.2.1.From the denition of length of curves B.2.8


n−1∑i=1

d(γ(ti), γ(ti+1)) (2.2.27)

But the partition a = t1 < b = t2 is part of the partitions of which thesupremum is taken so:

l(γ) ≥ d(γ(a), γ(b)). (2.2.28)

12

2.3 Shortest paths and geodesics 2 METRIC GEOMETRY

a):b): Choose ε > 0 and a partition P = t0, . . . , tN of γ such that:

l(γ)−N∑j=1

d(γ(tj−1), γ(tj) < ε (2.2.29)

Now consider:

Σ2 =

N∑j=1

d(γi(tj−1), γi(tj). (2.2.30)

Choose i such that:d(γi(tj), γ(tj)) <

ε

N(2.2.31)

for every tj ∈ P and denote:

Σ1 =

N∑j=1

d(γ(tj−1), γ(tj). (2.2.32)

By the triangle inequality:

d(γ(tj−1), γ(tj)) ≤ d(γ(tj−1), γi(tj−1)) + d(γi(tj−1), γi(tj)) + d(γi(tj), γ(tj))

so we get:

|d(γ(tj−1), γ(tj))− d(γi(tj−1), γi(tj))| ≤

≤ d(γ(tj−1), γi(tj−1)) + d(γ(tj), γi(tj)) ≤

≤ 2ε/N

This gives:

l(γ) < Σ1 + ε ≤ Σ2 + ε+ frac(2N)εN ≤ l(γi) + 3ε (2.2.33)

and since ε was chosen arbitrarily:

limi→∞

inf l(γi) ≥ l(γ).

2.3 Shortest paths and geodesics

Now the ground work is done to be able to show an important result from func-tional analysis which is here worked out for the special case of curves. Theconcept of compactness will henceforth be very important and for those readerswhich are not familiar with this it is recommended reading Appendix B.2 andstill a short look at this appendix is recommended for the more advanced readerin order to see how the dierent kind of compactness are dened in this thesis.

13


Theorem 2.3.1 (Arzela-Ascoli theorem for curves). [BBI01] Given a compactmetric space. Any sequence of curves which have uniformly bounded lengths hasan uniformly converging subsequence.

Remark 2.3.1. A sequence is bounded if every element in the sequence can bebounded by the same ball.

Proof. From Proposition 2.2.3 there exists constant speed parametrizations ofγi on the interval I[0,1]. Because the length of γi is uniformly bounded and wehave constant speed, there exists a c <∞ such that:

d(γi(t1), γi(t2)) ≤ l(γ(t1), γ(t2)) ≤ c|t1 − t2| (2.3.1)

for every i ∈ I and t1, t2 ∈ I[a,b].

Let S = tj be a countable dense subset of I[0,1]. From the Bolzano-WeierstrassTheorem B.2.2, there is a subsequence γni of γi such that for each j ∈ N, thesequence γni(tj) converges. Now the goal is to show that γni converges. Toavoid double indices call γni = γ′i.

The sequence γ′i(t) is a Cauchy sequence for all t ∈ I[0,1] because: Given ε > 0,choose tj ∈ S such that d(t− tj) < ε and an N ∈ N such that:

d(γ′i(t− tj), γ′k(tj)) < ε (2.3.2)

for every i, k > N . This gives that:

d(γ′i(t), γ′k(t)) ≤

≤ d(γ′i(t), γ′i(tj)) + d(γ′i(tj), γ

′k(tj)) + d(γ′k(tj), γ

′k(t)) ≤

≤ 3ε.

Because γ′i(t) is a Cauchy sequence we can dene

γ(t) = limj→∞

γ′j(tj). (2.3.3)

Using (2.3.1) gives:

d(γ(t1), γ(t2)) ≤ c|t1 − t2|, (2.3.4)

(2.3.4) gives that γ(t) is Lipschitz continuous and thus continuous due to The-orem A.0.4.

What is left to show now is that γ′i converges uniformly to γ: Given ε > 0,choose N > 4c · ε and let M be such that:

d(γ(k/N), γ′i(k/N)) <ε

2, for all k = 0, 1, 2, . . . , N and i > M. (2.3.5)

14


These choices are possible due to γ′i converging to γ pointwise.

Now using the results in (2.3.4) gives for all t ∈ I[0,1] and

k/N < t < (k + 1)/N : (2.3.6)

d(γ(t), γ′i(k/N)) ≤

≤ d(γ(t), γ′(k/N)) + d(γ′(k/N), γ′i(k/N)) ≤

≤ c|t− k/N |+ ε2 + c|t− k/N | ≤

≤ ε4 + ε

2 + ε4 ≤ ε

for every i > M . Thus the subsequence γ′i converges uniformly.

Here comes the rst result in this thesis which directly deals with shortest paths,in this case that converging sequences of shortest paths are shortest paths andthis result is surprisingly easy to show.

Proposition 2.3.1. [BBI01] Given a sequence γi of shortest paths in alength space (X, d) which converges to a path γ. Then γ is a shortest path.

Proof. Since the endpoints of γi converges to the endpoints a, b of γ and l(γi)equals the distance between the endpoints, the length

l(γi)→ d(a, b). (2.3.7)

Using the semi-continuity of length in Proposition 2.2.4 gives the following:

l(γ) ≤ limi→∞

l(γi) = d(a, b) (2.3.8)

But d(a, b) is the distance of the shortest path between a and b so γ is a shortestpath.

The following proposition which states that in compact metric spaces there ex-ist shortest paths between points which has rectiable curve between them isan important result. The condition that the space needs to be compact is astrong condition which can be weakened as will be seen later and the result ofProposition 2.3.2 still holds.

Proposition 2.3.2. [BBI01] Given a compact metric space (X, d). Leta, b ∈ X be points such that there exists a rectiable curve between a and b.Then there exists a shortest path between a and b.

15


Proof. Let linf be the inmum of lengths of rectiable curves between a and b.There exists a sequence of such curves γi such that l(γi)→ linf . From Arzela-Ascoli Theorem 2.3.1 there exists a converging subsequence γni of γi suchthat γni will satisfy that:

l(γni)→ linf . (2.3.9)

Let γni converge to a curve γ which will then have the end points a, b andusing 2.2.4 gives:

l(γ) ≤ limi→∞

l(γni) = linf (2.3.10)

and thus we have that l(γ) = linf and γ is a shortest path.

Now to the problem of getting a stronger variant of Proposition 2.3.2. One wayto obtain such results are to dene boundedly compactness and later use thesespaces.

Denition 2.3.1 (Boundedly compact). Given a metric space (X, d). Thespace X is boundedly compact if every closed and bounded subset of X is com-pact.

Example 2.3.1 (Boundedly compact). The space En is a boundedly compactspace. This very fact is proven in the Heine-Borel theorem B.2.1.

Example 2.3.2 (Boundedly compact). The space Rn with the discrete metric:

d(x, y) =

0, if x = y

1, if x 6= y(2.3.11)

and the standard topology is not boundedly compact. Choose a subset X ⊆ Rnsuch that X contains innitely many points of (Rn, d). The open sets in thistopology are all subsets of Rn. Choose the points of X as a covering of X , thenclearly it does not exists a nite subcovering of X and thus X is not compact.The subset X is bounded due to every set is bounded by a ball with radius r ≥ 1and it is closed. This can be realized by choosing x ∈ X and the neighborhoodx. This neighborhood does not contain any other points in X and therefore it isnot an accumulation point. Thus X does not contain any accumulation pointsand hence contain all of its accumulation points.

The following corollary is a stronger form of Proposition 2.3.2 where the condi-tion of compact space is changed to the weaker condition of boundedly compactspace.

Corollary 2.3.1. [BBI01] Given a boundedly compact space (X, d). For everyx, y ∈ X such that there exists a rectiable curve between them there exists ashortest path.

16


Proof. Given points x, y ∈ X and let l be the length of the curve between them.Considering the closed ball Bd(x, l), this is a compact set because Bd(x, l) isclosed, bounded and X is a boundedly compact space. Using Proposition 2.3.2on the ball shows that there exists a shortest path between x and y.

Corollary 2.3.1 can be improved further by using local compactness denedbelow instead of boundedly compactness. The goal now is to prove that com-pleteness and local compactness implies boundedly compactness and boundedlycompact can be interchanged with complete and locally compact in Corollary2.3.1.

Denition 2.3.2 (Locally compact). Given a topological space X. The spaceX is locally compact if for every x ∈ X there exists a compact neighborhood of x.

Remark 2.3.2. Note that in the literature there are several dierent and non-equivalent denitions of local compactness. Among others, a common denitionis that every point instead of a compact neighborhood has a neighborhood whoseclosure is compact (precompact). These denitions are not equivalent generallybut they are in Hausdor spaces and this is true for all other common denitionsof local compactness. The denition used in this thesis is implied by the seconddenition given in this remark and this denition could instead have been usedin this thesis almost without any changes in the proofs.

Example 2.3.3 (Locally compact). The space En is locally compact. This iseasily seen by choosing x ∈ En and then choosing the compact neighborhoodBd(x, r) of x.

Example 2.3.4 (Locally compact). The space

[0, 0, 0] ∪ [x, y, z] : x > 0 (2.3.12)

is not locally compact. For the point [0, 0, 0], there is no compact neighborhood.

Next to show that complete and locally compact implies boundedly compact wegeneralize the notion of sequences with the concept of nets and use nets to showthe implication.

Denition 2.3.3 (Net). Given a metric space (X, d). A subset Y ⊂ X is calledan ε-net of X if for every x ∈ X, d(x, y) < ε for some y ∈ Y .

Example 2.3.5 (Net). Let the metric space (X, d) be the the closed ball X =Bd(0, r + ε) ⊂ En with the euclidean metric. An ε-net to X is given by:

Y = Bd(0, r′), (2.3.13)

where r < r′ < r + ε.

17


Example 2.3.6 (Net). Given the metric space (R, d), where d is the Euclideanmetric. The subset Q ⊂ R is an ε-net of R for an arbitrary chosen ε > 0.

Before showing that that complete and local compactness implies bounded com-pactness the following lemma is needed.

Lemma 2.3.1. Given a closed set X in a complete metric space which is com-pact.

Then there exists a nite ε-net of X for every ε > 0.

Proof. Fix ε > 0 and x1 ∈ X. Now choose x2 ∈ X such that:

d(x1, x2) ≥ ε. (2.3.14)

Continue choosing x3, x4, . . . , xj+1 such that:

d(xi, xj+1) ≥ ε for every i = 3, 4, . . . , j. (2.3.15)

Now every x1, x2, . . . , xj+1 are of distance ε or more from each other but thisconstruction can not continue innitely because using Theorem B.2.3 gives thatX is sequentially compact and thus every sequence in X has a convergent subse-quence with limit inX. This gives a contradiction because every x1, x2, . . . , xj+1

are of distance ε or more from each other and thus for some j, x1, x2, . . . , xjthere is a pair xk, xl such that

d(xk, xl) < ε. (2.3.16)

Now choosing xj+1 ∈ X will give:

d(xj+1, xi) < ε for some i = 1, 2, . . . , j. (2.3.17)

Continue choosing xj+2, xj+3, . . . , xj+h until:

d(xk, xl) < ε for every pair xk, xl. (2.3.18)

This can be done by choosing xj+i in the following way: Let

r = maximin d(xi, xj) : for j = 1, 2, . . . , j + 1 and j 6= i . (2.3.19)

Then take the xi with the minimum index such that:

min d(xi, xj) : for j = 1, 2, . . . , j + 1 and j 6= i = r (2.3.20)

and denote as xk. Finally choose xj+i ∈ Bd(xk, r). Now the nite set xii=1,2,...,j+h

is an ε net.

18


Proposition 2.3.3. [BBI01] Given a complete locally compact length space(X, d). Then every closed ball in X is compact.

Remark 2.3.3. Note that Proposition 2.3.3 implies that the complete locallycompact length space is boundedly compact because every closed and boundedset Y ⊂ X can be contained in a closed, hence compact ball and using LemmaB.2.2 gives that Y is compact.

Proof. Choose x ∈ X and note that if the closed ball Bd(x, r) is compact, so is

Bd(x, r′) for r′ < r (2.3.21)

because Bd(x, r′) is closed, a subset of a compact set and hence compact fromLemma B.2.2.

LetR = sup

r > 0 : Bd(x, r) is compact

. (2.3.22)

Because X is locally compact there exists such a R > 0 and assume that R <∞.For convenience denote Bd(x,R) as B.

Since B is a closed set in a complete space it is sucient to show that thereexists a nite ε-net for every ε > 0, for B to be compact from Lemma 2.3.1.From the construction of B, ε < R can be assumed. Now consider the closedball:

Bd(x,R− ε/3) (2.3.23)

denoted as B′. From the construction of R the ball is compact and hence byLemma 2.3.1 it contains a nite ε-net denoted as E. Choose y ∈ B, this gives:

d(y, w) ≤ ε/3 for some w ∈ B′ (2.3.24)

because X is a length space. Furthermore

d(w, z) < ε/3 for some z ∈ E. (2.3.25)

This gives that d(y, w) < ε and hence B is closed.

For every y ∈ B there exists a compact neighborhood Uy. Choose a nitecollection Uyy∈Y that covers B which is possible due to Theorem B.2.4. Nowthe union

U =⋃y∈Y

Uy (2.3.26)

is a compact neighborhood for all elements in B. Construct a set U ′ such thatB is an ε-net of U ′ and U ′ is contained in U . Because X is a length space theball:

U ′ = Bd(x,R+ ε) (2.3.27)

19


is such a set for some ε > 0. Then

Bd(x,R+ ε) (2.3.28)

is a closed subset of a compact set U and hence compact from Lemma B.2.2.This contradicts the choice of R and hence R =∞ which give that every closedball in X is compact.

Now when it is shown that complete and local compactness implies boundedlyit is nally possible to strengthen the statement in Corollary 2.3.1 with the fol-lowing theorem.

Theorem 2.3.2. [BBI01] Given a complete locally compact length space (X, d).Then for every x, y ∈ X such that d(x, y) < ∞, there exists a shortest path γfrom x to y.

Proof. From Proposition 2.3.3, (X, d) is a boundedly compact space and usingCorollary 2.3.1 gives that there exists a shortest path between x and y ifd(x, y) <∞ (rectiable).

The above theorem is a very important result in this thesis and in fact the keyresult in this thesis the Hopf-Rinow-Cohn-Vossen Theorem 2.3.3 is the equiva-lence between the condition in Theorem 2.3.2 and three other conditions.

An important generalization of shortest paths are geodesics. The dierencebetween these are informally that geodesics are paths which only locally areshortest paths. The term historically comes from the science of geodesy whichexamines and measures distance, size and shape on the earth. There the termstood for shortest paths, i.e. great circles on the earth but was later generalizedto dierent geometries where it comes to be the "straight lines" in that geom-etry. For example in the general relativity the "straight lines" in the curvedspace time are the paths of which particles move inuenced only of gravitationand thus these paths are the geodesics in that geometry.

Denition 2.3.4 (Geodesic). Given a length space (X, d). A curve γ : I→ Xis a geodesic if for every x ∈ I, there exists an interval I[a,b] ⊆ I such that I[a,b]contains a neighborhood of x and with the property that the restriction of γ onI[a,b], γ|I[a,b] is a shortest path from γ(a) to γ(b).

Example 2.3.7 (Geodesic). Given the length space (S2, dI), where S2 is the2-dimensional unit sphere in R3. Let the curve

γ : I→ S2 (2.3.29)

be a line segment of a great circle (circle on S2 with radius 1). Then γ is ageodesic. This can easily be seen by choosing x ∈ S2 and then let the intervalI[a,b] be such that:

d(γ(a), γ(b)) ≤ π. (2.3.30)

20


Then γ|I[a,b] is a shortest path from γ(a) to γ(b) (see Figure 4).

Figure 4: A geodesic γ

Remark 2.3.4. From Example 2.3.7 it is clearly visible that it is not true thatevery geodesic is a shortest path. If γ is chosen such that there exists an intervalI[a,b] in the above example with

d(γ(a), γ(b)) > π. (2.3.31)

Then γ|I[a,b] is not a shortest path.

Earlier when the concept of shortest paths was dened, it was restricted tocurves of the form γ : I[a,b] → X. To generalize the notion for shortest paths tobe dened on non-closed intervals. The curve γ : I→ X is a shortest path or aminimal geodesic if for every interval:

I[a,b] ⊂ I, (2.3.32)

the restriction γ|I[a,b] is a shortest path, i.e. satisfying Denition 2.2.2.

21


Left to show now is only the main theorem of this chapter and also of the entirethesis, the Hopf-Rinow-Cohn-Vossen theorem.

Theorem 2.3.3 (Hopf-Rinow-Cohn-Vossen theorem). [BBI01] Given a locallycompact length space (X, d). Then the four following statements are equivalent:

a) X is a complete space

b) X is a boundedly compact space

c) Every naturally parametrized geodesic γ : I[0,a) → X is extendable to a con-tinuous pathγ′ : I[0,a] → X.

d) There exists a point x ∈ X such that every shortest path γ : I[0,a) → X,where γ(0) = x is extendable to a continuous path γ′ : I[0,a] → X.

Remark 2.3.5. Using Theorem 2.3.2 gives that all these four conditions impliesthat for every x, y ∈ X such that d(x, y) <∞ there is a shortest path betweenx and y if X is a locally compact length space.

Proof. We will show this theorem in the following order: b) =⇒ a), a) =⇒c), c) =⇒ d) and d) =⇒ b).

b) =⇒ a): Assume X is a boundedly compact space.

Because X is boundedly compact, every closed ball in X is compact. Nowchoose a Cauchy-sequence xk with limit:

limk→∞

xk = a. (2.3.33)

For some N ∈ N, the closure of the set xkk>N , denoted by K is contained

in a closed and hence compact ball Bd(xN , r). Since K is closed it contain itsaccumulation points and since a is an accumulation point to K,

a ∈ Bd(xN , r) (2.3.34)

and hence a ∈ X.

a) =⇒ c): Let γ : I[0,a) → X be a naturally parametrized geodesic. Then:

l(γ(c), γ(d) = d− c (2.3.35)

Choose a converging sequence:

ai ∈ I[0,a) such that 0 ≤ ai < a (2.3.36)

and converging to a. Letγ′i : I[0,ai] → X (2.3.37)

22


be such that γ′k is the same curve as the restriction γ|I[0,ak]. Due to (2.3.35), γ′i

is a Cauchy sequence and thus it has a limit γ′ : I[0,a] → X which is continuous.

c) =⇒ d): Assume that every geodesic γ : I[0,a) → X is extendable to a contin-uous path γ′ : I[0,a] → X.

If γ : I[0,a) → X is a shortest path, then

γ(0) = x for some x ∈ X. (2.3.38)

Because every shortest path is a geodesic, the curve γ is extendable to a curveγ′ : I[0,a] → X.

d) =⇒ b): Assume there exists a point x ∈ X such that every shortest pathγ : I[0,a) → X, where γ(0) = x is extendable to a continuous path γ′ : I[0,a] → X.

The proof of this follows the general scheme of the proof of proposition 2.3.3.BecauseX is locally compact, a suciently small closed ballBd(x, r) is compact.Let:

R = supr : Bd(x, r) is compact

(2.3.39)

and assume R <∞.

First prove that the open ball Bd(x,R) is precompact (its closure is compact)which will give that Bd(x,R) is compact. From Theorem B.2.3 it is sucientto show that every sequence xi contains a converging subsequence (note thatthe limit does not necessary belong to the ball). Let ri = d(x, xi) and assume:

limi→∞

ri = R (2.3.40)

which is possible because otherwise xi will be contained in a smaller compactball and that ball will have a converging subsequence.

Let γi : I[0,ri] → X be a shortest path from x to xi. To see that such a pathexists see the proof of Corollary 2.3.1. Now choose a subsequence of γi suchthat the restriction to the interval I[0,r1] is converging. Continue choosing fromthis subsequence such that the restriction to I[0,r2] is converging and so on.Then choosing the nth element of the nth subsequence (Cantor diagonalizationprocess) produces a sequence of paths γni such that for every t ∈ I[0,R) thesequence γni(t) converges in X and the endpoints of γni is a subsequencexni of xi.

Letting:γ(t) = lim γni(t) (2.3.41)

gives that γ is a shortest path from Proposition 2.3.1. From the assumptionthere is an extended continuous path γ′ : I[0,a] → X and the endpoints of theconverging curves γni converges to γ′(R). This gives that the subsequencexni of xi converges to γ′(R) and hence Bd(x,R) is compact.

23


Now show that a ball:Bd(x,R+ ε) (2.3.42)

for some ε is compact. Using that X is locally compact gives that there is acompact ball Bd(y, r(y)) around y for everyy ∈ Bd(x,R). Choose a nite covering of Bd(x,R) by the balls:

Bd(y, r(y)). (2.3.43)

Then the union of the closed balls U is a compact set and there is a ball:

Bd(x,R+ ε) (2.3.44)

for some ε such that:

Bd(x,R) ⊂ Bd(x,R+ ε) ⊂ U. (2.3.45)

Because Bd(x,R+ ε) is a closed subset of a compact set the Lemma B.2.2 givesthat Bd(x,R+ ε) is compact which contradicts the choice of R, hence R = ∞and X is boundedly compact.

In the two following chapters Riemannian geometry and Finsler geometry thetheory from this chapter will be applied. The Riemannian geometry with theright metric will be a length space and thus much of the theory from this chapterwill be applicable. In the Finsler geometry a problem arises because "choosing"the right "metric" will not give a length space because the chosen function isnot a metric but a semi metric. This creates some problems because the Hopf-Rinow-Cohn-Vossen theorem is not readily applicable but still it is possible toobtain a result corresponding to the Hopf-Rinow-Cohn-Vossen theorem in theFinsler geometry.

24

3 RIEMANNIAN GEOMETRY

3 Riemannian geometry

In 1854 Bernard Riemann held a famous lecture "Über die Hypothesen welcheder Geometrie zu Grunde liegen " where he lay the foundation of Riemanniangeometry by among other things generalizing the dierential geometry in R3

and lay the foundations of the theory of manifolds. Already during the rstpart of the 19th century a special case of a Riemannian geometry, the hyper-bolic geometry, was studied independently by the mathematicians Lobachevsky,Gauss and Bolyai.

The Riemannian geometry is a geometry characterized by a mathematical ob-ject called a manifold which locally resembles a subset of the n-dimensionalEuclidean space and a Riemannian metric which denes concepts like angles,volumes, length of curves, etc. Together these object constitutes a Riemannianmanifold which is the object of focus in Riemannian geometry.

A real world example of what a Riemannian geometry is and why they are use-ful is the length of the path a hiker traverses in a mountainous region. Usingthe Euclidean geometry the path traveled is "independent" of if the path goesthrough rough terrain and thus is very tiresome or not. Instead if this would bea Riemannian geometry the Riemannian metric would govern the length of thepaths by "penalizing" paths with rough terrain by adjusting the length pathby not only the length in the Euclidean sense but also how tiresome the pathis. Two possible interpretations of the length of these paths are either the timetraveled or the energy used by the hiker.

The content of this chapter is a classical approach of Riemannian geometry byintroducing the foundations of manifold theory, dening Riemannian manifoldsand the Riemannian connection in order to derive the Christoel symbols andthe geodesic equation. Then there is a section when the Riemannian geometryis handled as a metric space where a metric is dened such that a length spacearises and the Hopf-Rinow-Cohn-Vossen theorem is applicable. For a more thor-ough exposure of Riemannian geometry and the analysis of manifolds the books[GHL04] and [Küh02] are two good starting points where most of the theory inthis chapter can be found.

3.1 Manifolds

The theory of manifolds has been developed during the later half of the 19thcentury and as already stated Riemann was one of the early pioneers in thissubject. One informal example of manifolds which is important both for under-standing of where the notion originated from and how they work is the surfaceof a sphere. In R3 this object is a 2-dimensional abstract manifold which willbe seen later. Why it is a 2-dimensional abstract manifold is because by usingsubsets of R2, called charts, it is possible to represent the surface of the sphere.Think of these charts as an atlas covering the surface of the world and from thisanalogy a collection of charts is named an atlas.

Why the theory of manifolds is so useful is mainly because the theory for Rn isalready well-developed. Complicated objects which are hard to study directlythen behaves mostly like a subset of Rn when treated as a manifold and for

25

3.1 Manifolds 3 RIEMANNIAN GEOMETRY

example the results of calculus are mostly applicable.

In the rst section of this chapter the foundations of the theory of manifolds willbe dened. Abstract manifolds (dierentiable manifolds) will be used to denetangent spaces, tangents, the tangent bundle and then the tangent bundle isused to dene vector elds. For those acquainted with dierential geometrymost of these terms will be familiar and only generalized to abstract manifolds.

Before dening abstract manifold, submanifolds of Rn will be dened which willlater be used to to dene submanifolds of abstract manifolds.

Denition 3.1.1 (Dierential map). Given a function:

f =[f1 ... fm

]: Rn → Rm (3.1.1)

where f is Ck (k times continuously dierentiable). Then the dierential mapof f at p is given by:

Dfp =

∂f1

∂x1 (p) ... ∂f1

∂xn (p)...

...∂fm

∂x1 (p) ... ∂fm

∂xn (p)

(3.1.2)

Example 3.1.1 (Dierential map). The function f : R2 → R, given byf(x, y) = x2y, has the dierential map at p = [p1, p2] given by:

Dfp = [2p1 p2, p21]. (3.1.3)

The dierential for this function is given by:

dfp = 2p1 p2 dp1 + p21 dp2. (3.1.4)

Denition 3.1.2 (Submersion). A function f : M → N is a submersion if itsdierential map is surjective for all p ∈M .

Example 3.1.2 (Submersion). The function f : R2 → R, given by:

f(x, y) = x+ y (3.1.5)

is a submersion. The dierential map:

Dfp : R2 → [1, 1] (3.1.6)

at p = [p1, p2] is given by: Dfp = [1, 1]. To show that Dfp is surjective, takethe only element [1, 1] in the codomain and show there exists an element v ∈ R2

which is mapped on [1, 1], but this is satised for every element v ∈ R2 and thusDfp is a submersion.

26


Example 3.1.3 (Submersion). The function f : R→ R, given by f(x) = x3 isnot a submersion. The dierential map:

Dfp : R→ R (3.1.7)

at p is given by: Dfp = [3p2]. For the element −1 ∈ R there is no x in thedomain such that 3x2 = −1 and thus Dfp is not a submersion.

Denition 3.1.3 (Submanifolds of Euclidean spaces). A set:

M ⊂ Rn+m (3.1.8)

is an n-dimensional submanifold of class Ck of Rn+m if, for any p ∈ M , thereexists a neighborhood (B.1.9 on page 83) U of p in Rn+m and a Ck submersionf : U → Rm where

U ∩M = f−1(0). (3.1.9)

Example 3.1.4 (Submanifold). Consider the object:

M =x = [x1, x2, x3] ∈ R3 : f(x) = 4x2

1 + 2x22 + x2

3 − 4 = 0

(3.1.10)

This is a 2-dimensional submanifold of class C∞ of R3. The dierential is givenby:

dfx = 8x1 dx1 + 4x2 dx2 + 2x3 dx3 (3.1.11)

The function f dened in (3.1.10) is a submersion around every point in Mdue to the dierential map spanning R on M . Choose p ∈ M and let U be theneighborhood

R3 \Bd(0, ε), (3.1.12)

where Bd(0, ε) is a small ball such that:

Bd(0, ε) ∩M = ∅. (3.1.13)

ThenU ∩M = f−1(0). (3.1.14)

Example 3.1.5 (Submanifold). Consider the object:

M =x = [x1, x2, x3] ∈ R3 : f(x) = 4x2

1 + 2x22 + x2

3 = 0

(3.1.15)

This object is not a 2-dimensional submanifold of class C∞ of R3. In factM = 0 so this should not be surprising. Furthermore the function f denedin (3.1.15) is not a submersion around every point in M due to the dierentialmap not spanning R at 0 ∈ R3.

27


Before dening abstract manifolds, two properties for functions are needed,homeomorphism and dieomorphism. Homeomorphic functions are the coun-terpart to isomorphisms for topological spaces and thus such mappings preservesthe topological properties and the two spaces are essentially equivalent from atopological point of view.

Denition 3.1.4 (Homeomorphism). A function f : X → Y is a homeomor-phism between two topological spaces if the following is satised:

a) f is a bijection

b) f is continuous

c) f−1 is continuous

Example 3.1.6 (Homeomorphism). Considering the function:

f : (R, σ)→ (R, σ), (3.1.16)

where σ is the discrete topology (see denition B.1.4) and f is given by:

f(x) = 5x+ 3. (3.1.17)

f is obviously a bijection and to show that f is continuous, choose an arbitraryV ∈ σ. For f to be continuous, f−1(V ) has to be an open set in σ but this istrivial due to σ consisting of every subset of R. The inverse is

f−1(x) =1

5x− 3

5(3.1.18)

which is a continuous function using the same logic as for f , so f is a homeo-morphism.


f : (X,σ1)→ (Y, σ2), (3.1.19)

where σ1, σ2 are the discrete topologies of X, Y respectively,

X = −1, 0, 1 , Y = 0, 1, 2 (3.1.20)

and f is given by f(x) = x+ 1. f is obviously a bijection and f is continuous.The inverse is f−1(x) = x − 1. This function is continuous so f is a homeo-morphism.


f : (X,σ1)→ (Y, σ2), (3.1.21)

28


where σ1, σ2 are the discrete topologies of X, Y respectively,

X = −1, 0, 1 , Y = −1, 0, 1, 2 (3.1.22)

and f is given by f(x) = x + 1. Here f is not a bijection which was show inExample E.0.18. Choosing −1 ∈ σ2, now f−1(−1) = −2 is not an open set inσ1 so f is not continuous. The element −1 ∈ Y was not mapped on by anyelement in X so therefore the inverse to f does not exists.

A dieomorphism is closely related to homeomorphisms. Instead of only de-manding f and f−1 being continuous, they need to be continuously dieren-tiable and thus f also preserves the topological properties. The continuousdierentiability property will later give the dierentiable structure needed inthe abstract manifold.

Denition 3.1.5 (Dieomorphism). Given two manifoldsX and Y . A functionf : X → Y is a Ck dieomorphism if:

a) f is a bijective function (Bijection)

b) f is k-times continuously dierentiable (Dierentiable)

c) f−1 is k-times continuously dierentiable (Dierentiable)

Example 3.1.9 (Dieomorphism). Given the function f : R → R, where f isgiven by:

f−1(x) =1

5x− 3

5. (3.1.23)

This function is clearly bijective and f, f−1 are smooth functions so f is a C∞

dieomorphism.

The important step when creating an abstract manifold of a set is constructingan appropriate atlas. An atlas consists of an open covering of the set, togetherwith a collection of functions which connect every element of the covering witha subset of Rn and each function is a homeomorphism which preserve the topo-logical properties in the original set to the new collection of subsets of Rn.Furthermore some constraints in the choice of the homeomorphisms are neededin order to get the dierentiability of a manifold. Formalizing this gives thebelow denition.

Denition 3.1.6 (Atlas). Given a Hausdor space X. A Ck atlas is given by:

a) An open covering, Uii∈I of X, that is:

X ⊆ ∪i∈IUi. (3.1.24)

29


b) A family of homeomorphisms fi : Ui → Ωi (where Ωi ⊂ Rn), such that:for all index i, j ∈ I,

fj f−1i |fi(Ui∩Uj) (3.1.25)

is a Ck dieomorphism from fi(Ui ∩ Uj) onto fj(Ui ∩ Uj) (see Figure 5).

Figure 5: An atlas

Example 3.1.10 (Atlas). Considering the Hausdor space R with the standardtopology. The covering U1 = R together with the function f1 : U1 → R, given byf1(x) = x is a C∞ atlas. This can be seen by noticing that in fact U1 is triviallya covering of R, f1 is a homeomorphism with inverse

f−11 (x) = x (3.1.26)

satisfying b) in Denition 3.1.6. Here,

f1 f−11 = f(x) = x (3.1.27)

which is a C∞ dieomorphism from f1(U1) onto f1(U1).

30


Example 3.1.11 (Atlas). Considering the Hausdor space R3 with the standardtopology and the Euclidean metric F.0.12. The unit sphere S2 is a 2-dimensionalobject satisfying:

S2 =

[x, y, z] ∈ R3|x2 + y2 + z2 = 1

(3.1.28)

Constructing an atlas for S2 is possible with six charts. Considering the cover-ings Ui:

U1 =

[x, y, z] ∈ R3|x2 + y2 + z2 = 1 and z > 0

U2 =

[x, y, z] ∈ R3|x2 + y2 + z2 = 1 and z < 0

U3 =

[x, y, z] ∈ R3|x2 + y2 + z2 = 1 and x > 0

U4 =

[x, y, z] ∈ R3|x2 + y2 + z2 = 1 and x < 0

U5 =

[x, y, z] ∈ R3|x2 + y2 + z2 = 1 and y > 0

U6 =

[x, y, z] ∈ R3|x2 + y2 + z2 = 1 and y < 0

and the homeomorphism fi : Ui → D2:

f1(x, y, z) = [x, y]f2(x, y, z) = [x, y]f3(x, y, z) = [x, z]f4(x, y, z) = [x, z]f5(x, y, z) = [y, z]f6(x, y, z) = [y, z]

which maps the half-spheres on the unit disc D2.

The open covering6⋃i=1

Ui (3.1.29)

covers S2 where for example U1 is the upper semi-sphere in Figure 6 and thefunctions corresponding to the open covering are also homeomorphisms. For fito be a homeomorphism it needs to be a bijection. Here every element in Uimaps exactly one element of D2, every element in D2 was mapped on by exactlyone element in Ui so fi is a bijection which yields that f−1 exists. The functionsf and f−1 are continuous so f is a homeomorphism.

Here,fj f−1

i |fi(Ui∩Uj) (3.1.30)

is clearly well-dened and goes from fi(Ui ∩Uj) to fj(Ui ∩Uj). For example leti = 1 and j = 3, U1 ∩ U3 is the quarter-sphere dened by x ≥ 0 and z ≥ 0 (seeFigure 6). Then

f3 f−11 |f1(U1∩U3) (3.1.31)

has the domain f1(U1 ∩ U3) and

f−11 (f1(U1 ∩ U3)) = U1 ∩ U3. (3.1.32)

31


−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1−1

−0.5

0

0.5

1

Figure 6: A unit sphere

Because of this f3(U1 ∩ U3) is the corresponding codomain.

Now checking this for all pairs of i, j following the same scheme. For this to bean atlas

fj f−1i |fi(Ui∩Uj) (3.1.33)

need to be a dieomorphism. Due to fi being a homeomorphism, fjf−1i |fi(Ui∩Uj)

needs to be a bijection. Furthermore fj f−1i |fi(Ui∩Uj) and its inverse are smooth

functions so the pair Ui and fi is a C∞ atlas to S2.

Example 3.1.12 (Atlas). Given the unit circle S1. By a similar reasoning asfor S2 in Example 3.1.11, an atlas for S1 can be created by:

U1 =

[x, y] ∈ R2|x2 + y2 = 1 and x > 0

U2 =

[x, y] ∈ R2|x2 + y2 = 1 and x < 0

U3 =

[x, y] ∈ R2|x2 + y2 = 1 and y > 0

U4 =

[x, y] ∈ R2|x2 + y2 = 1 and y < 0

and the homeomorphism fi : Ui → I[0,1] where I[0,1] is the closed interval be-tween 0,1:

f1(x, y) = yf2(x, y) = yf3(x, y) = xf4(x, y) = x

Using the same logic as for S2 the pair Ui and fi are a C∞ atlas to S1.

32


Denition 3.1.7 (Maximal atlas). Given a Hausdor space X. Two Ck atlasesof X are equivalent if the union of the atlases is a Ck atlas of X. A maximalatlas of X is an equivalence class of equivalent atlases of X.

Remark 3.1.1. The equivalence class which denes a maximal atlas is also knownas a dierentiable structure.

When a maximal atlas is constructed to a set the resulting pair is an abstractmanifold and the denition is formalized below.

Denition 3.1.8 (Abstract manifold). An n-dimensional abstract Ck manifoldis a collection of a Hausdor space M and a maximal Ck atlas of M .

Remark 3.1.2. When referring to an abstract manifold, one element of the equiv-alence class will be given and letting that element represent the equivalence class.

Example 3.1.13 (Abstract manifold). The set R together with the C∞ atlasin Example 3.1.10, (U1, f1) is a 1-dimensional abstract C∞ manifold.

Example 3.1.14 (Abstract manifold). The set S2 together with the C∞ atlasin Example 3.1.11 is a 2-dimensional abstract C∞ manifold.

As mentioned in the introduction to this section, the atlas consists of a collec-tion of charts. Each chart can be thought of as a page in a "paper-version" ofan atlas. Such a construction is given by an open set from the covering with itscorresponding function which together forms a chart.

Denition 3.1.9 (Chart). For a given index i ∈ I, a pair (Ui, fi) , whereUi ∈M and fi satisfying Denition 3.1.6 are a chart of M .

Example 3.1.15 (Chart). The pair (U1, f1) in Example 3.1.10 is a chart of R.

Example 3.1.16 (Chart). Every (Ui, fi) in Example 3.1.11 is a chart for S2.

Early on in this chapter submanifolds of Rn was dened in Denition 3.1.3.Using these object a submanifold to an abstract manifold can be created bychoosing a subset of the manifold such that there exists, for any point in thesubset, a chart (f, U) covering that point and such that f(U) ∩Rn is submani-fold of an Euclidean space. This is formalized below.

Denition 3.1.10 (Abstract submanifold). Given a smooth n-dimensionalmanifold M . A subset N ⊂ M is an m-dimensional abstract submanifold ofM if there exists for any p ∈ N a chart (U, f) of M around p so f(U ∩N) is anm-dimensional submanifold of f(U) ∩ Rn.

33


Example 3.1.17 (Abstract submanifold). Consider the upper half-sphere ofS2 given by U1 in Example 3.1.11. Using that Example, there is always one ormore charts which includes a p ∈ U1. Choose one such chart, U, f. Clearlyf(U ∩U1) have to be a submanifold of f(U)∩R2 if U1 is an abstract submanifoldof S. The set f(U) ∩ R2 will be the 2-dimensional unit disc D2.

From here on unless otherwise stated an abstract manifold M will be assumedsmooth and n-dimensional.

When abstract manifolds are dened, the dierential structure can be used tocreate tangent vectors and tangent spaces to a curve. A problem arises becausedening the tangent at a point γ(p) ∈M as γ′(p) is not directly possible becausedierentiating is not dened. This is not a problem for submanifolds of Rn, forwhich the denition is straightforward.

Denition 3.1.11 (Tangent in submanifold). Given an n-dimensional subman-ifold M of Rn+k, p ∈M and x ∈ Rn+k. Then x is a tangent of M at p if thereis a curve γ on M such that γ(0) = p and γ′(0) = x

Example 3.1.18 (Tangent in submanifold). Given the manifold S2 in Example3.1.11 given by equation (3.1.28). This sphere is also a submanifold which canbe seen by using the same method as was used in Example 3.1.4. Consideringthe curve:

γ(t) =

x = sin(t), t ∈ I[−π,π]

y = 0, t ∈ I[−π,π]

z = cos(t), t ∈ I[−π,π]

(3.1.34)

This curve is on the submanifold S2 and γ(0) = [0, 0, 1] = p. Furthermore:

γ′(0) = [0, 0, 0] = x (3.1.35)

so x is a tangent at p to the submanifold S2.

The tangent spaces to manifolds are intuitively the "directional derivatives" ata point and if the tangents are dened the tangent space is dened as followed.

Denition 3.1.12 (Tangent space in submanifold). The set of tangents (tan-gent vectors) to M at p is called the tangent space of M at p, denoted TpM .

Example 3.1.19 (Tangent space in submanifold). Once again considering thesphere S2 in Example 3.1.11. The tangent space at a point p, TpS is given by:

[a, b, 0] | ∀a, b ∈ R (3.1.36)

which is the plane z = 1.

This can be seen by realizing that every curve γ on S2 passing through p has atangent vector of the form [a, b, 0] because the derivative of z at p will always be 0.

34


In this thesis two equivalent denitions of tangents of abstract manifolds will begiven. The rst denition is given below, which essentially says that a tangentis a tangent to a curve lying on the manifold.

Denition 3.1.13 (Tangent in abstract manifold). Given an abstract mani-fold M and a point p ∈ M . A tangent vector of M at p is an equivalence class(C.0.15 on page 106) of curves γ : I0 →M , where I0 is an interval containing 0and γ(0) = p. The equivalence relation (∼) is dened by:

γ0 ∼ γ1 ⇐⇒ Given chart (U, f) around p, the following holds:

(f γ1)′(0) = (f γ0)′(0) (3.1.37)

This denition is easy to understand but poses two problems. Firstly, the prop-erty of being a tangent vector to a manifold is not clearly independent of thechoice of chart. Secondly, it is not possible to chose a single principal represen-tative for an equivalence class of a tangent vector.

Example 3.1.20 (Tangent in abstract manifold). Given the sphere, S2 in Ex-ample 3.1.11. In Example 3.1.18 a tangent in the submanifold S2 was found.Other such curves γ are given by:

γ(t) =

x = sin(t) , t ∈ I[a,b]y = 0 , t ∈ I[a,b]z = cos(t) , t ∈ I[a,b]

(3.1.38)

where −π ≤ a < 0 and 0 < b ≤ π. All these curves belong to the same equiva-lence class, but these are not the only members for which the equivalence relationis fullled.

As for the tangent space of submanifolds, the tangent space of an abstract man-ifold at a point is simply the set of tangent vectors which in this case is a set ofequivalence classes.

Denition 3.1.14 (Tangent space in abstract manifold). The set of tangents(tangent vectors) toM at p is called the tangent space ofM at p, denoted TpM .

Example 3.1.21 (Tangent space in abstract manifold). The tangent space ofS2 at p = [0, 0, 1] is the set of equivalence classes for which the relation is ful-lled at p. Some of these curves gives rise to equivalence class given in Example3.1.20. Furthermore every pair a, b in Example 3.1.19 will generate a curvewhich will belong to one of the equivalence classes and every equivalence classhas a curve from Example 3.1.19.

The next object of interest is the tangent bundle. As the name indicates, thetangent bundle is informally the space of all tangent spaces for a manifold. Thisobject is important in the Riemann geometry and as will be seen later, many

35


−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1−100

−50

0

50

100

Figure 7: A tangent bundle

objects in the Riemannian and Finsler geometry are stated in terms of tangentbundles.

Denition 3.1.15 (Tangent bundle). Let the tangent bundle TM of M be thedisjoint union of the tangent spaces to M for all p ∈M .

Remark 3.1.3. The meaning of the word disjoint union above is that no twotangent spaces can have common vectors among them. For this to be possiblemore dimensions are needed. For an n-dimensional manifold the dimension ofthe tangent bundle is 2n.

Example 3.1.22 (Tangent bundle). Given the manifold S1 in Example 3.1.12.The tangent bundle of this object can be visualized by the Cartesian productS1 × R. Geometrically this could be interpreted by an innitely high cylinderconsisting of a circle centered at origo and radius 1 seen in Figure 7 if visualizedbeing of innite height.

Once the tangent bundle is dened vector elds comes naturally as the "di-rectional derivatives", as seen from the dierential geometry. Formalizing thiscomes naturally in the Riemannian geometry as:

Denition 3.1.16 (Vector eld). Given a manifold M . A vector eld in M iscalled a smooth section of the bundle TM , i.e. a smooth map X:

36


X : M → TM | for any p ∈M,X(p) ∈ TpM (3.1.39)

We will denote the set of all vector elds on M by Γ(TM) and X(p) by Xp

Example 3.1.23 (Vector eld). Given the sphere S1 in Example 3.1.12. Thesmooth map f : S1 → TS1 given by:

f(x, y) = g(x, y)[y,−x], (3.1.40)

where g is of the form g(x, y) : R2 → R (see Figure 8). Then f is a vector eldin S1 since f(x, y) is orthogonal (the inner product is zero) with [x, y]:

〈[x, y], g(x, y)[y,−x]〉 = x · g(x, y) · y − y · g(x, y) · x = 0 (3.1.41)

and hencef(x, y) ∈ T[x,y]S1. (3.1.42)

Another disadvantage of the denition now being used for tangents is that itbecomes unnecessarily hard to prove that tangent spaces are vector spaces. Be-cause of this another denition of tangent will be given which states that tangentvectors are derivations.

To dene a derivation, the concept of germs of functions is needed. The germsat a point can be thought of as the equivalence class of functions for a givenfunction class (C1, C2, continuous, etc.) for which every function is equal in asmall neighborhood of the point. More precisely stated as:

Denition 3.1.17 (Germs). Given a topological space X. The germs at p ∈ Xfor a given function class A (ex. continuous, Ck, smooth etc.) dened in aneighborhood of p is the equivalence class of functions f ∈ A satisfying theequivalence relation (∼), given by:

There is f1 : U → R and f2 : V → R such that there exists a neighborhood

W ⊂ (U ∩ V ) of p, (3.1.43)

such that f1(x) = f2(x) for every x ∈W (see Figure 9). This will denoted:

f1|W = f2|W (3.1.44)

henceforth.

37


Figure 8: A vector eld

38


Figure 9: Domain of germs

39


Notation 3.1.1 (Ckp (M)). The function space Ckp (M) are the germs of func-tions of the class Ck on the set M at a point p ∈M .

Example 3.1.24 (Germs). Given the topological space R with the standardtopology. Considering the functions f ≡ 0 and

g(x) =

0, t ∈ I[−1,1]

−x− 1, x < −1

x− 1, x > 1

. (3.1.45)

The neighborhood U = I(−1,1) in the standard topology yields for every x ∈ U ,f(x) = g(x). This gives that the functions f and g belongs to the same equiva-lence class (for example C∞) at for example p = 0 ∈ U .

Example 3.1.25 (Germs). Given the topological space R with the standardtopology. Considering the functions f(x) = x and g(x) = x3. These functionsdo not belong to the same germ because for every neighborhood U the functionsf and g do not coincide since f(x) = g(x) only at x ∈ −1, 0, 1.

Example 3.1.26 (Germs). Given the topological space R with the discrete topol-ogy. Considering the functions f(x) = x and g(x) = x3. These functions belongto the same germ in C0

p(M) at for example p = 0. Choose the the neighborhoodU = 0. Here f(U) = g(U), so they belong to the same equivalence class at p.

Denition 3.1.18 (Linear map). A linear map L : Ckp (M) → R satises forall a, b ∈ R and f, g ∈ Ckp (M):

L(af + bg) = a · L(f) + b · L(g) (3.1.46)

Example 3.1.27 (Linear map). The function L(x) = cx where c ∈ R is a linearmap.

Consider: L(af + bg) = c(af + bg) = acf + bcg = a · L(f) + b · L(g)

A derivation is simply dened as a linear map which satises the "product rule".Where the name comes from observing that every derivation can be written asa partial derivative as seen in the forthcoming Theorem 3.1.1.

Denition 3.1.19 (Derivation). A linear map L : Ckp (M)→ R is a derivationon Ckp (M) if for all f, g ∈ Ckp (M):

L(f · g) = f(p) · L(g) + g(p) · L(f) (3.1.47)

40


Example 3.1.28 (Derivation). Consider the linear map given by the directionalderivatives in the direction a = [a1, . . . , an]:

Daf =

n∑i=1

ai∂f

∂xi(3.1.48)

Calculating Da(fg) at p gives:

Da(fg) =n∑i=1

ai∂fg∂xi|p = f(p) ·

n∑i=1

ai∂g∂xi|p + g(p) ·

n∑i=1

ai∂f∂xi|p =

= f(p) ·Dag + g(p) ·Daf

From this follows that L is a derivation.

As already stated, an equivalent denition of tangent vectors are that they arederivations. More formally a tangent vector is dened as:

Denition 3.1.20 (Tangent in abstract manifold). A tangent vector L atp ∈M is a derivation on C∞p (M).

Proof. See for example [BJ82] for a proof of the equivalence.

The main reason for providing this alternate denition is its usefulness in calcu-lations which will be used now to prove that tangent spaces are vector spaces.Before stating and proving the main theorem, the following lemma is needed.

Lemma 3.1.1. [Küh02] Let X be a tangent vector and f a constant function,then

X(f) = 0. (3.1.49)

Proof. First assume f = 1. Using the denition of the derivation operation

(3.1.47):

X(1) = X(1 · 1) = 1 ·X(1) + 1 ·X(1) =

= 2 ·X(1) =⇒ X(1) = 0

In the general case f has a constant value c and using the denition of linearoperator (3.1.46):

X(c) = X(c · 1) = c ·X(1) = 0

41


Theorem 3.1.1. [Küh02] For an n-dimensional manifoldM , the tangent spaceat p is an n-dimensional vector space of Rn and TpM is spanned in any coordi-nate system x1, ..., xn for a given chart by:

∂

∂x1|p, ...,

∂

∂xn|p (3.1.50)

and for every tangent vector X at p:

X =

n∑i=1

X(xi)∂

∂xi|p (3.1.51)

Proof. Given a chart ϕ : U → V . Assume V = Bd(0, ε) which can be donewithout loosing generality and ϕ(p) = 0 so:

x1(p) = ... = xn(p) = 0. (3.1.52)

Let g : V → R be a smooth function and set f = g ϕ. Furthermore set:

gi(y) =

∫ 1

0

∂g

∂ei(t · y)dt (3.1.53)

where ei is the i:th component of standard basis in Rn which satises:

xi(p) = ei(ϕ(p)) (3.1.54)

and also due to g being smooth gi is smooth. Now the following calculations:

n∑i=1

∂g∂ei

(t · y) · d(tei)dt = (where d(tei)

dt = ei)

= ∂g∂t (t · y)

implies the following using above calculation and (3.1.53):

n∑i=1

gi(y) · ei =n∑i=1

∫ 1

0∂g∂ei

(t · y)dt · ei =

=∫ 1

0

n∑i=1

∂g∂ei

(t · y) · ei dt =1∫0

∂g∂t (t · y)dt = g(y)− g(0)

Now using the identities

f = g ϕ, fi = gi ϕ, xi = ei ϕ, (3.1.55)

the above expression and using ϕ(p) = 0 gives:

f(q)−f(p) =

n∑i=1

gi(ϕ(q)) · ei(ϕ(q)) +g(0)−g(ϕ(p)) =

n∑i=1

fi(q) ·xi(q) (3.1.56)

42


Taking derivatives gives:

∂f

∂xi|p= fi(p) (3.1.57)

Given a tangent vector X at p the following comes from the denition of linear

map 3.1.18, tangent vector 3.1.20, (3.1.56) and Lemma 3.1.1:

X(f) = X

(f(p) +

n∑i=1

fixi

)= X(f(p)) +X

(n∑i=1

fixi

)=

by tangent vector of constant and linear map

= 0 +n∑i=1

X(fixi) =

by the denition of derivation 3.1.20

=n∑i=1

(xi(p) ·X(fi) + fi(p)X(xi) =

by xi(p) = 0 and (3.1.57)

=n∑i=1

∂f∂xi|p·X(xi) =

(n∑i=1

X(xi) · ∂∂xi|p)

(f)

for every f ∈ C∞p (M) and left to show is that the vectors ∂∂xi|p are linearly

independent and that the tangent space is a vector space.

For TpM to be a vector space it needs to satisfy the conditions in DenitionD.0.4. These conditions are very repetitive to prove so only that if

X1, X2 ∈ TpM, X1 +X2 ∈ TpM (3.1.58)

will be proven here and the rest will be omitted.

X1(f) +X2(f) =

(n∑i=1

X1(xi) · ∂∂xi|p)

(f) +

(n∑i=1

X2(xi) · ∂∂xi|p)

(f) =

=

(n∑i=1

(X1(xi) +X2(xi)

)· ∂∂xi |p

)(f) ∈ TpM

43

3.2 Riemannian metrics 3 RIEMANNIAN GEOMETRY

To show that ∂∂xi|p is linearly independent lets consider the following equation:

∂

∂xi|p(xj) =

1 , i = j

0 , i 6= j(3.1.59)

For ∂∂xi|p to be linearly independent the linear combination of vectors have to

satisfy:

X(f) =

n∑i=1

X(xi)∂xj∂xi|p(f) = 0⇐⇒ X(xi) = 0 for every i (3.1.60)

Assume that X(xi) 6= 0 for some i. Then using 3.1.59 on

n∑i=1

X(xi)∂xj∂xi|p(x1) (3.1.61)

gives that X(x1) = 0 and continuing this for X(x2), X(x3),. . . , X(xn) gives acontradiction and ∂

∂xi|p is linearly independent.

3.2 Riemannian metrics

The second component in a Riemannian manifold is a Riemannian metric. With-out any extra structure, the length of curves in a manifold is undened but givena Riemannian metric, the length of curves and other geometric properties suchas angles will be dened. Once length of curves are dened, geodesics are alsodened. The Riemann metric denes at every point in the manifold an innerproduct such that it varies smoothly when moving on the manifold, this givesa family of inner products.

When dening Riemannian metrics, the concept of bilinear forms is needed. Abilinear form is essentially a map which is linear in each of its two argument.

Denition 3.2.1 (Bilinear form). Given a vector space X and a eld (C.0.18on page 108) of scalars, F . A bilinear form is a function B : X ×X → F thatsatises for all u, f, g ∈ X and c ∈ F :

a) B(u+ f, g) = B(u, g) +B(f, g)

b) B(u, f + g) = B(u, f) +B(u, g)

c) B(cf, g) = B(f, cg) = c ·B(f, g)

Example 3.2.1 (Bilinear form). Given a vector space X over the eld R. Thisvector space is equipped with an inner product 〈·, ·〉. As seen for u, f, g ∈ X andc ∈ R this function is a bilinear form.

a) 〈u+ f, g〉 = 〈u, g〉+ 〈f, g〉

b) 〈u, f + g〉 = 〈u, f〉+ 〈u, g〉

44


c) 〈cf, g〉 = c · 〈f, g〉 = 〈f, cg〉

Example 3.2.2 (Bilinear form). Given a vector space X over the eld C. Thisvector space is equipped with an inner product 〈·, ·〉. As seen for u, f, g ∈ X andc ∈ C this function is not a bilinear form.

a′) 〈u+ f, g〉 = 〈u, g〉+ 〈f, g〉

b′) 〈u, f + g〉 = 〈u, f〉+ 〈u, g〉

c′) 〈cf, g〉 = c · 〈f, g〉 = 〈f, cg〉

In Denition 3.2.1 of bilinear forms condition a) and b) are fullled but notc). A function which satises a) and b) and is conjugate linear as 〈·, ·〉 in thisexample is called a sesquilinear form.

The inner products will belongs to the space (L2(TpM ;R)) dened as:

Denition 3.2.2 ((L2(TpM ;R))). This space is dened by:

L2(TpM ;R) = f : TpM × TpM → R| f is bilinear (3.2.1)

L2(TpM ;R) has the basis:

dxi|p ⊗ dxj |p| i, j = 1, ..n (3.2.2)

Here dxi is a dual basis in the dual space: (TpM)∗

Where we dene:

dxi|p(∂

∂xj|p) = δij =

1 , if i = j

0 , if i 6= j(3.2.3)

The bilinear forms dxi|p ⊗ dxj |p are dened by:

(dxi|p ⊗ dxj |p)(

∂

∂xk|p, ∂

∂xl|p)

= δikδjl =

1 , if i = k and j = l

0 , otherwise(3.2.4)

Inserting the basis and coecients for the representation gives:

α =∑i,j

αijdxi ⊗ dxj (3.2.5)

which gives the following expression:

αij = α

(∂

∂xi,∂

∂xj

)(3.2.6)

45


Then a Riemannian metric is a collection of α's as in (3.2.5) which are symmet-ric, positive denite and the coecients αij are dierentiable.

Denition 3.2.3 (Riemannian metric). On a given manifoldM , a Riemannianmetric g is an association for points p ∈M :

p→ gp ∈ L2(TpM ;R) (3.2.7)

such that the following is satised for all x, y ∈ TpM :

a) gp(x, y) = gp(y, x) (Symmetry)

b) gp(x, x) > 0 for all x 6= 0 (Positive deniteness)

c) In every local representation (in every chart) the coecients gij

gp =∑i,j

gij(p)dxi|p ⊗ dxj |p (3.2.8)

are dierentiable functions. (Dierentiability)

Remark 3.2.1. The inner product dened by the Riemann metric g at a pointp ∈ TM is the gp from the above denition. If the condition of positive de-niteness in Denition 3.2.3 is replaced by the condition that if:

gp(x, x) = 0 =⇒ x = y, (3.2.9)

then g is a semi-Riemannian metric (or pseudo-Riemannian metric). A classicalexample of a semi-Riemannian metric is the Lorentzian metric from the generalrelativity theory.

Example 3.2.3 (Riemannian metric). Given an open set U ∈ Rn and let thebasis be given by e1, . . . , en. Then using ∂

∂xi= ei and (3.2.6)gives:

gij = gp

(ei, ej

)(3.2.10)

If gp is the Euclidean inner product then gij is the identity matrix:

gij =

1 0 . . . 00 1 . . . 0...

.... . .

...0 . . . 0 1

. (3.2.11)

Due to gp being an inner product a) and b) in the denition of Riemannianmetric 3.2.3 is fullled. For gp to be a Riemannian metric the coecients gijhas to be dierentiable functions which is excluded here but satised and hencethe function g(·, ·) = 〈·, ·〉 is a Riemannian metric on U .

46

3.3 Riemannian connections 3 RIEMANNIAN GEOMETRY

Now a Riemannian manifold is a manifold together with a Riemannian metric.This object has the sucient structure for further analysis, the dierentiabilityof the manifold make it possible to use calculus on complicated objects and theRiemann metric gives length of curves which give rise to geodesics which in turnare essential to this thesis.

Denition 3.2.4 (Riemannian manifold). A pair (M, g) is a Riemannian man-ifold if M is a manifold and g is a Riemannian metric

Example 3.2.4 (Riemannian manifold). The pair (U, g) where U is an openset, U ∈ Rn and g is the Riemannian metric from Example 3.2.3 is a Rieman-nian manifold.

An important question is if there exist Riemann metrics for a given manifoldand if so whether they are unique. As proven in Theorem 3.2.1, there exist in-deed Riemann metrics for any given manifold but they are generally not uniqueas seen by the following example.

Example 3.2.5 (Non-uniqueness of Riemannian metric). Given an open setU ∈ Rn. In Example 3.2.3 a Riemann metric to this manifold was given bythe Euclidean inner product. Another example of a Riemann metric for thismanifold is given by:

gij(x1, . . . , xn) = δij(1 + xki xkj ) (3.2.12)

where k is an even integer and is given in matrix form by:

gij =

1 + xk1 0 . . . 0

0 1 + xk2 . . . 0...

.... . .

...0 . . . 0 1 + xkn

. (3.2.13)

Theorem 3.2.1 (Existence). [GHL04] There exist at least one Riemannianmetric g on any given manifold M .

Proof. The proof is omitted here and can be found in [GHL04].

3.3 Riemannian connections

In Section 3.1 the problem of derivatives on abstract manifolds are dealt withfor the case of scalar functions by Denition 3.1.20 of tangent vectors. In thissection the notion of derivatives on abstract manifolds will be introduced bydening the derivative of vector elds with respect to a tangent vector whichwill give a tangent vector. One way to do this is by the Lie bracket which doesnot use the Riemannian metric and another is the Riemannian connection whichuses the Riemannian metric and is a generalization of the covariant derivative

47


which is the usual derivative of vector elds with respect to a tangent vectorin dierential geometry. As will be shown the Riemannian connection is in factunique for every Riemannian metric and using the Riemann connection, theChristoel symbols can be expressed which will then be used to express thegeodesic equation in local coordinates.

The Lie bracket or the Lie derivative is a measure of the non-commutativity ofthe derivatives of the vector elds and is dened as:

Denition 3.3.1 (Lie bracket). Given a Riemannian manifold (M, g) with vec-tor elds X,Y onM and a smooth function f : M → R. The Lie bracket [X,Y ]of X,Y is dened as:

[X,Y ](f) = X(Y (f))− Y (X(f)) (3.3.1)

and at a point p the Lie bracket is dened as:

[X,Y ]p(f) = Xp(Y (f))− Yp(X(f)) (3.3.2)

Remark 3.3.1. Note that in the denition of Lie bracket, the Riemannian met-ric g was not used so only the dierential structure is sucient for a denition.Also, the Lie bracket [X,Y ] is a vector eld which can be realized by remem-bering that the set of tangent vectors is a vector space.

Example 3.3.1 (Lie bracket). Given the vector elds X,Y , where

X = xy∂

∂x(3.3.3)

and

Y = (x+ y)∂

∂y(3.3.4)

and the smooth function f : S1 → R, where f(x, y) = x + y. Then the Lie

bracket is given by:

[X,Y ](f) = xy

∂

((x+y)

∂(x+y)∂y

)∂x − (x+ y)

∂

((xy)

∂(x+y)∂x

)∂y =

= xy − (x2 + xy) = −x2

Below follows six important properties of Lie brackets which will be used con-tinuously throughout this section when calculating with Lie brackets.

Proposition 3.3.1 (Properties of Lie brackets). [Küh02] Given a Riemannianmanifold M with vector elds X,Y, Z on M , constants a, b ∈ R and smoothfunctions f, h : M → R. The Lie bracket satises the following properties:

48


a) [aX + bY, Z] = a[X,Z] + b[Y,Z] (Linear in rst argument)

b) [X,Y ] = −[Y,X] (Anti-symetric)

c) [fX, hY ] = f · h · [X,Y ] + f ·X(h) · Y − h · Y (f) ·X

d)[X, [Y,Z]

]+[Y, [Z,X]

]+[Z, [X,Y ]

]= 0 (Jacobi identity)

e) [ ∂∂xi

, ∂∂xj

] = 0

for every chart with coordinates [x1, . . . , xn]

f)[ ∑

i

ξi∂∂xi

,∑j

ηj∂∂xj

]=∑i,j

(ξi∂ηj∂xi− ηi ∂ξj∂xi

)∂∂xj

(Representation in local coor-

dinates)

Proof.

a): [aX + bY, Z](f) =

= a ·X(Z(f)) + b · Y (Z(f))− Z(a ·X(f) + b · Y (f)) = (Z is a linear map)

= a ·X(Z(f))− a · Z(X(f)) + b · Y (Z(f))− b · Z(Y (f)) =

= a · [X,Z](f) + b · [Y, Z](f)

b): [X,Y ](f) = X(Y (f))− Y (X(f)) = −(Y (X(f))−X(Y (f))

)= −[Y,X](f)

c): [fX, hY ](θ) = f ·X(h·Y (θ))−h·Y (f ·X(θ)) = (using equation (3.1.47) in Denition 3.1.19)

= f ·X(h)Y (θ) + f · h ·X(Y (θ))− h · Y (f)(X(θ))− h · f · Y (X(θ)) =

= (f · h · [X,Y ] + f ·X(h) · Y − h · Y (f) ·X)(θ)

for every function θ : M → R .

d): Using:

[X, [Y,Z]

](f) = X

([Y, Z](f)

)−[Y, Z](X(f)) =

=

(X(Y (Z(f)

)−Z(Y (f)

))−(Y(Z(X(f))

)−Z(Y (X(f))

))

gives:

([X, [Y,Z]

]+[Y, [Z,X]

]+[Z, [X,Y ]

])(f) =

=

(X(Y (Z(f)

)−Z(Y (f)

))−(Y(Z(X(f))

)−Z(Y (X(f))

))+

+

(Y(Z(X(f)

)−X

(Z(f)

))−(Z(X(Y (f))

)−X

(Z(Y (f))

))+

+

(Z(X(Y (f)

)−Y(X(f)

))−(X(Y (Z(f))

)−Y(X(Z(f))

))=

= 0

e):

[∂∂xi

, ∂∂xj

](f) = ∂

∂xi

(∂∂xj

(f))− ∂∂xj

(∂∂xi

(f))=

= ∂2f∂xi∂xj

− ∂2f∂xj∂xi

= 0

49


f): Considering: [∑i

ξi∂

∂xi,∑i

ηi∂

∂xi

](xj) (3.3.5)

gives:[ ∑i

ξi∂∂xi

,∑i

ηi∂∂xi

](xj) =

∑i

ξi∂∂xi

(∑i

ηi∂xj∂xi

)−∑i

ηi∂∂xi

(∑i

ξi∂xj∂xi

)=

=∑i

ξi∂ηj∂xi−∑i

ηi∂ξj∂xi

Using (3.1.51) gives that:[∑i

ξi∂

∂xi,∑j

ηj∂

∂xj

]=∑i,j

(ξi∂ηj∂xi− ηi

∂ξj∂xi

)∂

∂xj(3.3.6)

The meaning of the term Riemannian connection (or Levi-Civita connection)lies in the "connection" between the tangent spaces which by denition are dis-joint. These tangent spaces at dierent points in the manifold are related bythe Riemannian connection as following:

Denition 3.3.2 (Riemannian connection). Given a Riemannian manifold (M, g).A Riemannian connection ∇ on (M, g) is a map:

∇ : TM × TM → TM (3.3.7)

which associate two vector elds X and Y with a vector eld ∇XY such thatthe following is satised for a smooth function f : M → R:

a) ∇X1+X2Y = ∇X1Y +∇X2Y (additive in the subscript)

b) ∇fXY = f · ∇XY (linear in the subscript)

c) ∇X(Y1 + Y2) = ∇XY1 +∇XY2 (additive in the argument)

d) ∇X(fY ) = f · ∇XY + (X(f)) · Y (product rule in the argument)

e) X(g(Y,Z)) = g(∇XY, Z)+g(Y,∇XZ) (relation with the Riemannian metric)

f) ∇XY −∇YX = [X,Y ] (torsion-free)

Example 3.3.2 (Riemann connection). Given the Riemannian manifold (Rn, g),where g is the Euclidean inner product. For vector elds:

X = [X1, . . . , Xn] =∑i

ai∂

∂xi, Y =

∑i

bi∂

∂xi, Z =

∑i

ci∂

∂xi, (3.3.8)

the directional derivative:

DXY =∑i

Xi∂Yi∂xi

∂

∂xi= ∇XY (3.3.9)

is a Riemannian connection on (Rn, g) which is shown below:

50


a): ∇X+ZY =∑i

(ai + ci)∂bi∂xi

=∑i

ai∂bi∂xi

+∑i

ci∂bi∂xi

= ∇XY +∇ZY

b): ∇fXY =∑i

fai∂bi∂xi

= f∑i

ai∂bi∂xi

= f · ∇XY

c): ∇X(Y + Z) =∑i

ai∂(bi+ci)∂xi

=∑i

ai∂bi∂xi

+∑i

ai∂ci∂xi

= ∇XY +∇XZ

d): ∇X(fY ) =∑i

ai∂(fbi)∂xi

= f ·∑i

ai∂bi∂xi

+∑i

ai∂f∂xi

bi = f · ∇XY + (X(f)) · Y

Properties e) and f) are omitted here but can be computed in a similar man-ner although more tedious.

As already stated, for a given Riemannian manifold, there exists an unique Rie-mannian connection which is formalized in the following theorem.

Theorem 3.3.1. [Küh02] [KN63] Given a Riemannian manifold (M, g). Thenthere exists an unique Riemannian connection ∇ on (M, g).

Proof. First prove the uniqueness of the connection ∇. For vector elds X,Y, Z,the following three equalities holds true using the property of the relation withthe Riemannian metric in the denition of the Riemannian connection 3.3.2 anddenoting gp(·, ·) = 〈·, ·〉.

X〈Y, Z〉 = 〈∇XY,Z〉+ 〈Y,∇XZ〉 (3.3.10)

Y 〈X,Z〉 = 〈∇YX,Z〉+ 〈X,∇Y Z〉 (3.3.11)

− Z〈X,Y 〉 = −〈∇ZX,Y 〉 − 〈X,∇ZY 〉 (3.3.12)

Adding (3.3.10), (3.3.11) and (3.3.12) gives:

X〈Y, Z〉+ Y 〈X,Z〉 − Z〈X,Y 〉 =

= 〈Y,∇XZ−∇ZX〉+〈X,∇Y Z−∇ZY 〉+〈Z,∇XY+∇YX〉 = (∇XY−∇YX = [X,Y ])

= 〈Y, [X,Z]〉+ 〈X, [Y,Z]〉+ 〈Z, 2∇YX + [Y,X]〉

Rearranging terms in the above equation gives the Koszul formula below:

2〈∇XY, Z〉 = X〈Y,Z〉+Y 〈X,Z〉−Z〈X,Y 〉−〈Y, [X,Z]〉−〈X, [Y, Z]〉−〈Z, [Y,X]〉(3.3.13)

Given Z, the right-hand side of (3.3.13) is uniquely determined. Assume thatthere exists an U 6= ∇XY such that for all Z:

〈∇XY,Z〉 = 〈U,Z〉. (3.3.14)

Then〈∇XY − U,Z〉 = 0 (3.3.15)

51


and this is especially true for

Z = ∇XY − U (3.3.16)

which gives that:

〈∇XY − U,∇XY − U〉 = 0, (3.3.17)

using that M is Hausdor and 〈·, ·〉 is an inner product together with equation(3.3.17) gives ∇XY = U and hence ∇XY is unique.

Now to show the existence of ∇, dene it as satisfying the Koszul formula(3.3.13) for every X,Y, Z ∈ Γ(TM).

Remain to show is that ∇XY is well-dened and thus:

〈∇XY |p, Zp〉 (3.3.18)

only depends on Zp.

For ∇ to be a Riemannian connection it has to satisfy condition a)-f) in De-nition 3.3.2.

a) Using (3.3.13) on 2〈∇X1+X2Y,Z〉 − 2〈∇X1

Y +∇X2Y,Z〉 gives:

2〈∇X1+X2Y,Z〉−2〈∇X1

Y +∇X2Y, Z〉 = (X1 +X2)〈Y, Z〉+Y 〈X1 +X2, Z〉−

−Z〈X1 +X2, Y 〉−〈Y, [X1 +X2, Z]〉−〈X1 +X2, [Y,Z]〉−〈Z, [Y,X1 +X2]〉−

− (X1〈Y,Z〉+X2〈Y,Z〉+ Y 〈(X1, Z〉+ Y 〈(X2, Z〉 −Z〈X1, Y 〉 −Z〈X2, Y 〉 −

−〈Y, [X1, Z]〉−〈Y, [X2, Z]〉−〈X1, [Y,Z]〉−〈X2, [Y,Z]〉−〈Z, [Y,X1]〉−〈Z, [Y,X2]〉 =

= 0

by using the bilinearity of the inner product, the anti-symmetry of the Lie

brackets and collecting terms.

b) Using (3.3.13) on 2〈∇fXY,Z〉 − 2〈f∇XY,Z〉 gives:

(fX〈Y,Z〉+Y 〈fX,Z〉−Z〈fX, Y 〉−〈Y, [fX,Z]〉−〈fX, [Y, Z]〉−〈Z, [Y, fX]〉)−

−(fX〈Y,Z〉+Y 〈X,Z〉−fZ〈X,Y 〉−f〈Y, [X,Z]〉−f〈X, [Y,Z]〉−f〈Z, [Y,X]〉) =

= [ Using the product rule for Lie brackets and the bilinearity of inner products ] =

= Y f〈X,Z〉−Zf〈X,Y 〉+fZ〈X,Y 〉−fY 〈X,Z〉−〈Y, f [X,Z]〉−〈Y,−(Zf)X〉+

52


+ f〈Y, [X,Z]〉 − 〈Z,−f [X,Y ]〉 − 〈Z,−(Y f)X〉+ f〈Z, [Y,X]〉 =

= [ Using the product rule on Zf〈X,Y 〉 = (Zf)〈X,Y 〉+fZ〈X,Y 〉 and Y f〈X,Y 〉 =

= (Y f)〈X,Z〉+ fY 〈X,Z〉]

= (Y f)〈X,Z〉+ fY 〈X,Z〉 − (Zf)〈X,Y 〉 − fZ〈X,Y 〉 − fY 〈X,Z〉+

+ fZ〈X,Y 〉+ (Zf)〈X,Z〉 − (Y f)〈X,Z〉 =

= 0.

The rest of the conditions c) − f) are done in a similar manner and are

skipped in this thesis.

Now the goal with the rest of this section is to express an equivalent denitionof geodesics in the Riemannian geometry to and use this denition to nd anequation in local coordinates for the geodesic equation. The Christoel symbolsare expressed in terms of the Riemann metric and the Riemann connection.The new denition of geodesics gives an equation a curve has to satisfy to be ageodesic. When expressing this in local coordinates, the Christoel symbols areexpressed and more generally the Christoel symbols arises when dealing withRiemann connections in local coordinates.

Denition 3.3.3 (Christoel symbols). The Christoel symbols of the rst kindΓij,k is dened as:

Γij,k =1

2(− ∂

∂xkgij +

∂

∂xjgik +

∂

∂xigjk) (3.3.19)

and the Christoel symbols of the second kind Γmij is dened as:

Γmij =∑k

Γij,kgkm,∇ ∂

∂xi

∂

∂xj=∑k

Γkij∂

∂xk(3.3.20)

we then have:

∇ ∂∂xi

∂

∂xj=∑k

Γkij∂

∂xk(3.3.21)

The following proposition gives a formula for the Riemann connection in localcoordinates.

Proposition 3.3.2. Given vector elds:

X =∑i

ξi∂

∂xi, Y =

∑j

ηj∂

∂xj(3.3.22)

53


in local coordinates. Then ∇XY in local coordinates is given by:

∇XY =∑k

(∑i

ξi∂ηk∂xi

+∑i,j

Γkijξiηj

)∂

∂xk. (3.3.23)

Proof. Using the properties a)-d) in Denition 3.3.2 on ∇XY gives:

∇XY = ∇∑iξi

∂∂xi

(∑j

ηj∂∂xj

)= ξ1∇ ∂

∂x1

(∑j

ηj∂∂j

)+ . . .+ ξn∇ ∂

∂xn

(∑j

ηj∂∂j

)=

= ξ1η1∇ ∂∂x1

∂∂x1

+ ξ1∂η1∂x1

∂∂x1

+ . . .+ ξ1ηn∇ ∂∂x1

∂∂xn

+ ξ1∂ηn∂x1

∂∂xn

+

+ ξ2η1∇ ∂∂x2

∂∂x1

+ ξ2∂η1∂x2

∂∂x1

+ . . .+ ξnηn∇ ∂∂xn

∂∂xn

+ ξn∂ηn∂xn

∂∂xn

=

=∑k

(∑i

ξi∂ηk∂xi

+∑i,j

Γkijξiηj

)∂∂xk

An alternative equivalent denition of geodesics 2.3.4 in Chapter 2 for Rieman-nian geometry is given below.

Denition 3.3.4 (Geodesic in Riemannian geometry). Given a Riemannianmanifold (M, g) and a curve γ. The curve γ is a geodesic in a Riemanniangeometry if

∇ dγdt

dγ

dt= λ

dγ

dt(3.3.24)

for some function λ : M → R and if γ is a natural parametrization (see Denition2.2.4),

∇ dγdt

dγ

dt= 0 (3.3.25)

To see why this equation make sense at least in En as a denition for geodesics,consider the equation:

∇ dγdt

dγ

dt=dγ2

dt2. (3.3.26)

This is the vector of acceleration for a particle moving along a curve γ and ifthe particle moves without the inuence of acceleration, it will follow a straightline which are the geodesics in En and without acceleration (3.3.24) is reducedto a geodesic equation. If instead γ is a curve on S2. Then the vector of ac-celeration is given by: ∇ dγ

dt

dγdt . The geodesics will be the path particles follow

without the inuence of acceleration which will be the great circles which againcoincide with what was discovered in Chapter 2.

Proof. (Equivalence of the denitions) The proof of the equivalence of the twodenitions is omitted here but can be found in [Jos11].

54

3.4 Geodesics 3 RIEMANNIAN GEOMETRY

Using Proposition 3.3.2 together with (3.3.25) gives a system of dierential equa-tions for the geodesic equation in local coordinates.

Proposition 3.3.3. [GHL04] Given a Riemannian manifold (M, g). Then anaturally parametrized curve is a geodesic if it is a solution to the following dif-ferential equations in local coordinates:

d2xkdt2

+∑i,j

Γkijdxidt

dxjdt

= 0, (3.3.27)

where xi are the coordinates for γ.

Proof. See [GHL04] for a proof of this proposition.

3.4 Geodesics

In Chapter 2, which concerns metric geometry, several results about when short-est paths exist in length spaces was given. To use these results for a Riemannianmanifold, a suitable metric in the Riemannian geometry is needed in order fora Riemannian manifold to be a length space. This can and will in fact be donelater in this subsection and as will be seen the choice of metric is logical andstraightforward.

As stated in the introduction to this chapter, the Riemannian metric is usedwhen dening the length of a curve by "weighting" the dierent parts of thepath of the curve.

Denition 3.4.1 (Length of curve). Given a Riemannian manifold (M, g) anda continuously dierentiable curve γ : I[a,b]→M on M . The length of the curvel(γ) is given by:

l(γ) =

b∫a

√gγ(t)(γ′(t), γ′(t))dt (3.4.1)

Example 3.4.1 (Length of curve). Given a Riemannian manifold (M, g), whereg is the Euclidean inner product. The length of the curve:

γ(t) =

x = t, t ∈ I[0,1]

y = t, t ∈ I[0,1]

(3.4.2)

is given by:

l(γ) =1∫0

√gγ(t)(γ′(t), γ′(t))dt =

1∫0

√〈[1, 1], [1, 1]〉dt =

1∫0

√2dt =

√2.

The goal is to choose a metric such that the corresponding metric space is alength space. The metric then needs to coincide with the intrinsic metric from

55


Chapter 2 in order to be a length space. Because length of curves are onlydened for C1 curves, the inmum of every path of C1 curves between the givenpoints will coincide with the intrinsic metric because the manifolds are smooth.

Denition 3.4.2 (Metric on Riemannian manifold). A metric d on a connectedRiemannian manifold (M, g) is given by:

d(p1, p2) = inf l(Φ) (3.4.3)

where Φ is the set of C1 curves of the form γ : I[a,b] →M,γ(a) = p1 and γ(b) = p2.

Left to show for the above dened metric to be a length space is that it is indeeda metric and that the topology generated by this metric coincides with the usualtopology of the manifold.

Proposition 3.4.1. [GHL04] The object d dened in Denition 3.4.2 is ametric and the topology generated by the metric is the original topology of themanifold.

Proof. First prove that d is a metric. In order to be well-dened, d has to bedened for every pair x, y ∈M .

Choose x ∈ M and let Ux be the set of points joined to x by piecewise C1

curves. Then Ux is an open set and

U \ Ux =⋃y/∈Ux

Uy (3.4.4)

is hence an open set because it is an union of open sets. Now Ux is non-emptybecause x ∈ Ux and U \ Ux is the empty set, due to otherwise there wouldbe two disjoint open sets Ux, U \ Ux which covers M and hence M would bedisconnected. Then Ux = M and d is well-dened.

A metric satises:d(x, y) = d(y, x). (3.4.5)

Because length of curves is independent of backward and forward parametriza-tion of curves this is fullled. The triangle inequality,

d(x, y) ≤ d(x, z) + d(z, y) (3.4.6)

comes from the fact that the distance is the inmum of the length of the curvesand hence the length of a piece-wise C1 curve between x to y passing z cannotbe greater than d(x, y).

The last property d must satisfy is:

d(x, y) = 0⇐⇒ x = y. (3.4.7)

56


If x = y, then clearly d(x, y) = 0. Now prove the converse assertion. Assumex 6= y. Now there exists a chart (X, f) around x such that:

y /∈ X, f(x) = 0, f(X) = Bh(0, 1) (3.4.8)

because M is Hausdor and h is the metric induced by f on Bh(0, 1) by theRiemannian metric g i.e. the "Riemannian metric" on the submanifold X. Let‖·‖ denote the Euclidean norm in Rn. Using the compactness of Bh(0, 1

2 ) givesthat there exist λ, µ > 0 such that for a given p ∈ Bh(0, 1

2 ) and u ∈ TpM :

λ‖u‖2 ≤ hp(u, u) ≤ µ‖u‖2. (3.4.9)

Choosing a curve γ between x and y and calculating l(γ) (see [GHL04]) givesthat l(γ) > 0, which in turn gives: d(x, y) > 0 which is a contradiction andhence d is a metric.

What is left to prove is that the topologies coincide which is omitted here andcan be found in [GHL04].

Summarizing the above results. A connected Riemannian manifold togetherwith the metric dened in Denition 3.4.2 is a length space. Together with thefact from Proposition 3.4.1 that this metric induces the same topology as theoriginal topology of M gives that a connected Riemannian manifold is a lengthspace and the theory in Chapter 2 about length spaces is applicable. Rephras-ing the Hopf-Rinow-Cohn-Vossen Theorem 2.3.3 for Riemannian manifolds givesthe following:

Theorem 3.4.1. Given a locally compact theorem Riemannian manifold M .Then the following conditions are equivalent

a) M is a complete space.

b) M is a boundedly compact space.

and if the conditions a) and b) are fullled, for every x, y ∈ M such thatd(x, y) <∞ there is a shortest path between x and y

Proof. This theorem follows easily from Theorem 2.3.2, Hopf-Rinow-Cohn-VossenTheorem 2.3.3 and that M is a length space.

Example 3.4.2 (Existence of shortest paths). Given a Riemannian manifold(S2, g) with a Riemannian metric g. This manifold is locally compact, completeand there exist rectiable paths between every point. Then there exist shortestpaths (not necessarily unique) between every pair of points in S2.

57

4 FINSLER GEOMETRY

4 Finsler geometry

The Finsler geometry is a generalization of the Riemannian geometry. In theRiemannian geometry, the Riemannian metric induces an inner product but thefunction corresponding to the Riemann metric in the Finsler geometry inducessomething called a Minkowski norm instead of an inner product. One other waythe Minkowski norms dier is that they are functions on the tangent bundle andhence they have dependence of both the position in the manifold and the corre-sponding tangent space. Reusing the example of the hiker in the mountainousregion, used in the Riemannian geometry. In the Finsler geometry the length ofthe path the hiker traverses is a function of which Minkowski norm is used andhence the length of the path does not only depends on which places were visitedbut also in which direction the hiker traveled at the time, so for example a pathdownwards a mountain may be shorter than the same path traveled upwardsthe mountain in the Finsler geometry.

The Finsler geometry is named after Paul Finsler who studied it in his disserta-tion from 1918. The name was phrased by Élie Cartan who was also one of theearly pioneers in this subject in the rst part of the 20:th century. Among otherimportant contributers in this subject are: Ludwig Berwald, Vagner Busemann,Makoto Matsumoto and Shiing-Shen Chern. Most of the result in this chaptercan be found in [BCS00] and for those not familiar with dierential forms thereis a section in Appendix D covering most of what is needed as a prerequisiteabout dierential forms to understand this chapter and for those interested ina more thorough explanation of dierential forms the book [Car06] is recom-mended.

The content of this chapter is two sections covering the classical approach toFinsler geometry which follows the approach in Riemannian geometry closely bywhich introducing Finsler manifolds and the Chern connection which replacesthe Riemannian connection in the in Riemannian geometry. Then comes a sec-tion covering geodesics in the Finsler geometry where it will be shown that theapproach used to create a length space by introducing a suitable metric in theRiemannian geometry will fail in the Finsler geometry.

4.1 Finsler manifolds

The Finsler manifold is a generalization of the Riemannian manifold. As withthe Riemannian manifold, it is a manifold together with a function which decidesthe geometrical properties of the Finsler geometry such as angles and length ofcurves.

In this chapter let x denote a point in a manifold M and let y ∈ TxM if nototherwise specied. Furthermore let the map:

π : TM →M, π(x, y) = x (4.1.1)

be the natural projection (see Figure 10). By the object TM \ 0 means everyelement of TM except when y = 0.

59

4.1 Finsler manifolds 4 FINSLER GEOMETRY

Denition 4.1.1 (Finsler manifold). Given a manifoldM . A Finsler structureF is a function F : TM → R+ which satises for all x, y ∈ TM and real λ > 0:

a) F is smooth on TM \ 0 (Regularity)

b) F (x, λy) = λF (x, y) (Positive homogeneity)

c) The n× n matrix:

gij =

[1

2

∂2

∂yi∂yjF 2

](4.1.2)

is positive-denite at every point of TM \ 0

A pair (M,F ) is a Finsler manifold.

Remark 4.1.1. The function F in Denition 4.1.1 evaluated at a specic tangentspace is a Minkowski norm and as can be proven it is in fact a norm as seenin [BCS00].

Example 4.1.1 (Finsler manifold). Given a manifold M and a Riemannianmetric g. The function F , dened in a point x ∈M as:

F (x, y) =√gx(y, y) (4.1.3)

is a Finsler structure. Due to the smoothness of gij, the function F is a smoothfunction because F is positive on TM \ 0. Using that gx is an inner productyields that:

F (x, λy) =√gx(λy, λy) = λ

√gx(y, y) = λF (x, y) (4.1.4)

for λ > 0. The positive deniteness of[1

2

∂2

∂yi∂yjF 2

](4.1.5)

is due to the Riemannian metric gx at a point x being positive-denite.

Remark 4.1.2. A Finsler manifold which is also a Riemannian manifold is aRiemannian Finsler manifold .

Example 4.1.2 (Finsler manifold). For a given µ > 0 and y = [y1, y2]. Let theFinsler structure F (x, y) be independent of x and given by:

F (x, y) =

√√y4

1 + y42 + µ(y2

1 + y22) (4.1.6)

Then the pair (R2, F ) is a Finsler manifold. These kind of Finsler manifoldswhere F is independent of x are known as locally Minkowskian manifolds. The

60

4.2 Connections 4 FINSLER GEOMETRY

function F is a smooth function on TM \ 0 and calculating F (λy1, λy2) forλ > 0 gives:

F (λy1, λy2) =

√√(λy4

1) + (λy2)4 + µ((λy1)2 + (λy2)2) = λF (y1, y2) (4.1.7)

Using the denition and calculating gij gives:

[g11 g12

g21 g22

]=

λ+y21(y41+3y42)

(y41+y42)3/2−2y31y

32

(y41+y42)3/2

−2y31y32

(y41+y42)3/2λ+

y22(y42+3y41)

(y41+y42)3/2

. (4.1.8)

Observing that (4.1.8) is an Hermitian matrix and that the determinant:

det

([g11 g12

g21 g22

])> 0 (4.1.9)

if λ > 0 by Example C.1.1 and using Theorem C.1.1 gives that[g11 g12

g21 g22

]is

positive-denite.

From now on in this chapter a variation of the Einstein summation conventionwill be used, i.e. expressions of the type:

X =

n∑i=1

Xiei (4.1.10)

is written as X = Xiei with the Einstein summation convention. Vectors will

have superscript indices vi where i goes from 1 to n, basis elements ci will havesubscript indices and repeated indices in a expression in both superscript andsubscript means summing over that index. The indices of summation are as-sumed to be between 1 and n. This gives that a vector v can be written as:v = cie

i whereeiis a basis and ei is a vector.

For covectors (elements of the dual space) the indices are subscript and theelements of the basis are superscript indices. This gives that a covector w canbe written as: w = cie∗i where e∗i is a basis of the dual space and e∗i is acovector. For lowering and raising indices the object gij and its inverse gij isused respectively.

4.2 Connections

In the Riemannian geometry, the Riemannian connection was used to measurethe derivative of vector elds with respect to a tangent vector and thus thisgives a "connection" between dierent tangent spaces. In the Finsler geometryseveral dierent connections arises. In this thesis the following will be used:Non-linear connection, Ehresmann connection, linear connection and the Chernconnection. The Chern connection will have the same role as the Riemannianconnection in the Riemannian geometry and the goal of the theory of this chap-ter is to dene and prove that the Chern connection always exists and is uniquefor a given manifold. The meaning of the term connection can informally be

61


thought of as an object which can connect points in dierent charts which other-wise would not easily be compared. For a more formal treatment on connectionwith emphasize on Finsler geometry the book [Mat86] is recommended.

Before the Chern connection can be dened a lot of ground work is needed.Several abstract object and spaces will be presented below in order to ease thecalculation later on.

Denition 4.2.1 (Sphere bundle). Given a manifold (M). The sphere bundleSM is the set of points: [

x,y

‖y‖

]∈ TM \ 0

, (4.2.1)

where an element [x, y] is identied in the sphere bundle by:

[x, y‖y‖

].

Example 4.2.1 (Sphere bundle). Given the Riemannian Finsler manifold (S1, g),where g is the Euclidean inner product. Then TM is given by S1×R from Ex-ample 3.1.22 and hence SM is the set of points:

[x, y] : x ∈ S1, y =±[x2 − x1]

‖[x2 − x1]‖

(4.2.2)

for x = [x1, x2].

Figure 10: Tangent bundle

The method used for dening the sphere bundle can also been seen as treatingevery ray:

[x, λy] : λ > 0 (4.2.3)

62


as a single point and dene TxM in this point (see Figure 4.2.3) together withthe inner product:

gij(x, y)dxi ⊗ dxj . (4.2.4)

The point of this is that gij(x, y) is invariant under rescaling, y → λy for λ > 0,i.e.

gij(x, y) = gij(x, λy). (4.2.5)

This shows that while the dimension of the vector bundle TM is 2n the di-mension of SM is 2n − 1. Further on in this chapter other objects which areinvariant under rescaling will be considered to simplify the calculations.

The canonical projection map

p : SM →M,p(x, y) = x (4.2.6)

helps dene the pulled-back bundle p∗TM . Another pulled-back bundle is π∗TMon TM \0 which serves the same purpose as p∗TM does on SM . This meansthat π∗TM is a vector bundle over TM \ 0 and π∗TM at a specic point(x, y) is a copy of TxM .

On π∗TM there is a natural Riemannian metric

g = gijdxi ⊗ dxj (4.2.7)

which is called the fundamental tensor and another important tensor is the Car-tan tensor A dened by:

A = Aijkdxi ⊗ dxj ⊗ dxk, (4.2.8)

where

Aijk =F

2

∂gij∂yk

=F

4

∂3

∂yi∂yj∂ykF 2 (4.2.9)

is invariant under rescaling and nally the object:

Cijk =1

FAijk. (4.2.10)

In the literature this tensor is an alternative denition of the Cartan tensor butCijk is not invariant under rescaling.

As in the Riemannian geometry, Christoel symbol for the Finsler geometry canbe dened as:

Denition 4.2.2 (Christoel symbols). In the Finsler geometry the formalChristoel symbols of the second kind Γijk on TM \ 0 is given by:

Γijk = gis1

2(∂gsj∂xk

− ∂gjk∂xs

+∂gks∂xj

) (4.2.11)

63


Denition 4.2.3 (Non-linear connection). The object:

N ij = Γijky

k − CijkΓkrsyrys (4.2.12)

on TM \ 0 is the non-linear connection.

Because objects which are invariant under rescaling is preferable, introducingthe object:

N ij

F= Γijk`

k −AijkΓkjk`r`s, (4.2.13)

where `i = yi

F denes the distinguished section ` of π∗TM by:

` =yi

F

∂

∂xi= `i

∂

∂xi(4.2.14)

which gives an object which is invariant under rescaling and its dual analog onπ∗T ∗M is the Hilbert form ω given by:

ω =∂F

∂yidxi. (4.2.15)

Several of the following computations are excluded in this thesis but can befound in [Kaw56] and [Kik62].

A local change of variables xi = xi(x1, . . . , xn) forNijF gives:

Npq

F=∂xp

∂xi∂xj

∂xpN ij

F+∂xp

∂xi∂2xi

∂xq∂xs˜s. (4.2.16)

To see where the name non-linear connection comes from consider local coordi-nate changes on the coordinate basis ∂

∂xi and∂∂yi . The transformation on ∂

∂xi

is given by the slightly complicated expression:

∂

∂xp=∂xi

∂xp+

∂2xi

∂xp∂xqyq

∂

∂yi(4.2.17)

while the expression for variable changes on ∂∂yi is less complicated.

∂

∂xp=∂xi

∂xp∂

∂xi. (4.2.18)

and the local coordinate basisdxi, dxj

of the dual space of TM called the

cotangent space, is transformed by coordinate changes to:

dxp =∂xp

∂xidxi (4.2.19)

64


dyp =∂yp

∂yidyi +

∂2xp

∂xi∂xj(4.2.20)

where the expression for dyp is in this case more simple than that of dxp.

Now replace ∂∂xi by:

δ

δxj=

∂

∂xj−N i

j

∂

∂yi(4.2.21)

and dyi by:

δyi = dyi +N ij dx

j (4.2.22)

but this object is not invariant under rescaling so instead introduce the followingobject:

δyi

F=

1

Fdyi +N i

j dxj . (4.2.23)

Now note that the corresponding dual objects of δδxj ,

δyi

F are respectively dxj ,F ∂∂yi and that the above objects have easy transformations under coordinate

changes.

Now the following object on TM \ 0 is a Riemannian metric and known asthe Sasaki metric:

gijdxi ⊗ dxj + gij

δyi

F⊗ δyi

F. (4.2.24)

Using this metric, the space spanned by

δδxj

is the horizontal subspace and

the orthogonal space spanned byδyi

F

is the vertical subspace. These spaces

build up what is known as an Ehresmann connection and this connection ischaracterized by the quantities N i

j which is where the name non-linear connec-tion comes from.

When measuring the rate of change of a tensor eld Y in the direction X at apoint p it has to satisfy the product rule. Let Λ be such an operator and let itsatisfy the product in the following way.

ΛX` = (d`j)(X)∂

∂xj+ `jΛX

∂

∂xj, (4.2.25)

ΛXg = (dgij)(X)dxi ⊗ dxj ∂+gij(ΛXdx

i)⊗ dxj + gijdxi ⊗ (ΛXdx

j). (4.2.26)

and dene the covariant derivatives ΛX∂∂xj and ΛXdx

i as:

ΛX∂

∂xj= ωij(X)

∂

∂xi, (4.2.27)

ΛXdxi = −ωij(X)dxj . (4.2.28)

65

4.3 Geodesics 4 FINSLER GEOMETRY

In the Riemannian geometry, the Riemannian connection is a generalizationof the covariant derivative. Further generalizations of the covariate derivativeto the Finsler geometry gives the linear connections which are dened below.Comparing these connections to the Riemannian geometry, the linear connec-tions does not satisfy the torsion free condition which expressed in terms ofFinsler geometry is stated as:

d(dxi)− dxj ∧ ωij = −dxj ∧ ωij = 0 (4.2.29)

Denition 4.2.4 (Linear connection). The operator Λ on π∗TM which satisesthe equations (4.2.25)-(4.2.28) together with the following conditions for vectorelds X,Y , scalar functions f and constants λ is a linear connection.

a) ΛX(fY ) = (df)(X)Y + fΛXY

b) ΛX(Y1 + Y2) = ΛXY1 + ΛXY2

c) ΛλXY = λΛXY

d) ΛX1+X2Y = ΛX1

Y + ΛX2Y

For some examples of dierent linear connection and other kind of connection,the text book [Mat86] is recommended.

The linear connections are not expressed in terms of the Finsler structure. TheChern connection is a linear connection which imposes constraints such thatit both relates to the Finsler structure and is torsion free. By this the Chernconnection generalizes the Riemannian connection. In the following theoremthe Chern connection is dened and as stated, it exists and is unique for a givenFinsler manifold as is the case in the Riemannian geometry for Riemannianmanifolds and the Riemannian connection.

Theorem 4.2.1. [BCS00] Given a Finsler manifold (M,F ). There existsan unique linear connection called the Chern connection ∇ which satisfy thefollowing conditions:

a) d(dxi)− dxj ∧ ωij = −dxj ∧ ωij = 0 (Torsion freeness)

b) dgij − gkjωki − gikωkj = 2Aijsδys

F (Almost g-compatibility)

Proof. The proof of this theorem is omitted but can be found in either of [BCS00]and [Mat86].

4.3 Geodesics

As for geodesics in the Riemannian geometry, a similar approach in Finsler ge-ometry for geodesics is conducted to create an equation governing whether acurve is a geodesic or not and as such serving as an equivalent denition of a

66


geodesic curve in the Finsler geometry.

The denition of the length of a curve in the Finsler geometry follows the sameconcept as in the Riemannian geometry. This means using the Finsler structurefor "weighting" the dierent regions of the manifold. One big dierence betweenthe two geometries is that the length of a curve in the Finsler geometry is notindependent of the direction in which the path is traveled as is the case in theRiemannian geometry.

Denition 4.3.1 (Length of curve). Given a Finsler manifold (M,F ) and acontinuously dierentiable curve γ : I[a,b] →M on M . The length of the curve,denoted l(γ), is given by:

l(γ) =

b∫a

F (γ(t), γ′(t))dt. (4.3.1)

To dene the concept of geodesics in the Finsler geometry, the rst variation ofarc length is used. It is the derivative of the arc length of curves γ(t, u) whichwill be explained below.

Given a piecewise smooth curve γ(t), 0 ≤ t ≤ r with a partition:

0 = t1 < t2 < . . . < tk = r (4.3.2)

such that γ is smooth on every part of the partition. Then the length of γ (arclength) is given by:

l(γ) =

k∑i=2

ti∫ti−i

F (γ(t), γ′(t))dt. (4.3.3)

Now consider the rectangular domain:

= (t, u) : 0 ≤ t ≤ r,−ε < u < ε (4.3.4)

A piecewise smooth variation of γ(t) is a continuous map γ(t, u) from to themanifold M which satises that on each rectangle:

I[ti−1,ti] × I(−ε,ε) (4.3.5)

the map is smooth and that γ(t, 0) = γ(t).

For every xed value of u the mapping γ(t, u) give rise to a t-curve and similarlyif t is constant it is an u-curve. The length of a t-curve (denoted l(u)) for agiven variation γ(t, u) is given by:

l(u) =

k∑i=2

ti∫ti−1

F

(γ(t, u),

∂γ

∂t(t, u)

)dt. (4.3.6)

67


Now dierentiating (4.3.6) with respect to u gives the rst variation of arclength. To see how this is done and to get a more thorough explanation of thissubject see 5.1 in [BCS00].

Then an alternative equivalent denition to the general Denition 2.3.4 givenin Chapter 2 for a curve being geodesic in the Finsler geometry is given below.

Denition 4.3.2 (Geodesics in Finsler geometry). Given a Finsler manifold(M,F ) and a piecewise smooth curve γ : I[0,r] →M onM . Then γ is a geodesicif:

l′(0) = 0 (4.3.7)

for every piecewise smooth variation of γ with xed endpoints.

Proof. The equality of the general denition of geodesics and the specic de-nition in the Finsler geometry can be found in [BCS00]

For some examples of geodesics in the Finsler geometry and further discussionsof this topic see [BCS00].

In the Riemannian geometry the metric in Denition 3.4.2 gives a length spacefor a Riemannian manifold. Hopefully using this "metric" on Finsler manifoldswill also give a length space for Finsler manifolds but unfortunately this is notthe case due to the resulting "metric" is in fact not a metric.

Denition 4.3.3 ("Metric" on Finsler manifold). Let the function d on theFinsler manifold (M, g) be given by:

d(p1, p2) = inf l(Φ) (4.3.8)

where Φ is the set of continuous curves of the form γ : I[a,b] →M,γ(a) = p1 and γ(b) = p2.

The function d dened in Denition 4.3.3 satises the rst and third conditionin Denition A.0.9 for being a metric but not the second property of being ametric, namely:

d(x1, x2) = d(x2, x1). (4.3.9)

Example 4.3.1 (A Finsler manifold for which d is not a metric). Finsler man-ifolds whose Finsler structure is not absolute homogeneous:

F (x, λy) = |λ|F (x, y) for every λ ∈ R (4.3.10)

does not necessarily satisfy the symmetry condition for d being a metric. This

68


can be intuitively realized by observing that when moving in the opposite direc-tion of the path dening d(x1, x2), i.e the inmum length of paths between x2

and x1, in the point x, the value of the Finsler structure does not necessarilycoincide with the value of the Finsler structure at x when moving between x1

and x2 and hence the lengths of the two paths are not always equal.

The Finsler manifold (R2, F ) seen in Example 4.1.2 and dened as:

F (x, y) =

√√y4

1 + y42 + µ(y2

1 + y22) (4.3.11)

where y = [y1, y2]. This Finsler manifold is absolute homogeneous and hence dis a metric on (R2, F ).

Remark 4.3.1. A function which satisfy the rst and third condition of being ametric is called a semi-metric.

To show that d is a semi-metric the following lemma is used.

Lemma 4.3.1. [BCS00] Given a the Finsler manifold (M,F ). There existsat every point x ∈ M a local coordinate system φ : U → Rn with the followingproperties for an open set U where U is the closure of U and a constant c > 1:

a) U is compact, φ(x) = 0 and φ maps U dieomorphically on an open ball inRn.

b) There exists a constant c > 1 such that:

1

c|y| ≤ F (x, y) ≤ c|y| and F (x,−y) ≤ c2F (x, y) (4.3.12)

for every y = yi ∂∂xi∈ TpM and p ∈ U . Furthermore here, |y| =

√δijyiyj

where

δij =

0, if x 6= y

1, if x = y(4.3.13)

which is known as the Kronecker delta.

c) Given x1, x2 ∈ U . Then:

1

c|φ(x2)− φ(x1)| ≤ d(x1, x2) ≤ c|φ(x2)− φ(x1)|. (4.3.14)

d) Given x1, x2 ∈ U . Then:

1

c2d(x2, x1) ≤ d(x1, x2) ≤ c2d(x2, x1). (4.3.15)

69


Proof. a) Choose a local coordinate system φ : W → Rn where W ⊂ M andφ(x) = 0. Let B(r) be an open ball dened by:

B(r) =xi ∈ Rn : xi <

√δij

(4.3.16)

Take a r > 0 such that:ˆB(r) ⊂ φ(W ) (4.3.17)

Then the inverse image:φ−1(B(r)) (4.3.18)

is precompact and hence φ : W → Rn satisfy the rst condition of the lemma.

The rest of the conditions are omitted here but can be found in [Mat86].

Before proving the rst and third properties for being a metric for d, the "topol-ogy generated by d" will be proved to equal the topology of the manifold.

The balls which will generate the topology are the forward metric balls whichare dened below.

Denition 4.3.4 (Forward metric ball). Given a Finsler manifold (M,F ). Aforward metric ball B+(x, r) centered at x with radius r is given by:

B+(x, r) = p ∈M : d(x, p) < r . (4.3.19)

Remark 4.3.2. Note that the order of x and p is important in the denition offorward metric ball due to the non-symmetry of d.

Example 4.3.2 (Forward metric ball). Consider the Finsler structure denedby the Riemann metric in Example 3.2.3 which is is just the Euclidean innerproduct. Then the forward metric balls in the Finsler manifold (Rn), F ) are justthe open balls dened by the Euclidean metric.

Theorem 4.3.1. [BCS00] The topology of the Finsler geometry coincides withthe topology generated by the forward balls when seen as a metric space, i.e aset is open if around every point in the set there exists a forward ball containingthe point and being contained by the set.

Proof. The proof of this theorem is omitted here but can be found in [BCS00].

Now it is time to prove that d satises the rst and third condition of a metric.

Theorem 4.3.2. [BCS00] Given a Finsler manifold (M,F ). Then the functiond dened in Denition 4.3.3 satises for every x1, x2 ∈M :

70


a) d(x1, x2) = 0 ⇐⇒ x1 = x2.

b) d(x1, x3) ≤ d(x1, x2) + d(x2, x3).


Because d is not a metric, a Finsler metric is not necessarily a length space andhence the Hopf-Rinow-Cohn-Vossen Theorem 2.3.3 is not readily accessible. In-stead, a variant of this theorem will be developed and to do that, the Finslergeometry counter part to Cauchy sequences will be needed.

Denition 4.3.5 (Forward Cauchy sequence). Given a Finsler manifold (M,F ).A sequence xi ∈ M is a forward Cauchy sequence if, for every ε > 0, thereexists an N ∈ N such that:

N ≤ i < j =⇒ d(xi, xj) < ε. (4.3.20)

The reason Forward Cauchy sequences are needed are for the counterpart ofHopf-Rinow-Cohn-Vossen theorem in the Finsler geometry which naturally shouldstate something concerning complete spaces but due to the non-symmetry of the"metric" function, Cauchy sequences are not dened and hence Forward Cauchysequence will be used to dene "complete" spaces in Finsler geometry.

Denition 4.3.6 (Forward complete space). Given a Finsler manifold (M,F ).The Finsler manifold is a forward complete space if every Forward Cauchy se-quence converges in M .

The last thing needed to dene is the Finslerian counterpart to boundedly com-pact spaces which is not surprisingly given by:

Denition 4.3.7 (Forward boundedly compact space). Given a Finsler mani-fold (M,F ). A subset of the Finsler manifold is forward bounded if it is containedin a forward metric ball. The Finsler manifold is a Forward boundedly compactspace if every closed and forward bounded subset is closed.

Finally we can state and prove the Hopf-Rinow-Cohn-Vossen theorem in theFinsler geometry.

Theorem 4.3.3. [BCS00] Given a connected Finsler manifold (M,F ). Thenthe following statements are equivalent:

a) The space (M,F ) is forward complete.

b) The space (M,F ) is a forwardly bounded space.

71


and both these conditions gives that there exists a shortest path γ between everypoint x1, x2 ∈M .


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A METRIC SPACES

Appendices

The following appendices are meant to cover the important results and deni-tion which are not directly related to the subjects of this thesis but still neededin order to fully understand all the material covered in this thesis. One goal ofthese appendices is to be self-contained so no outside resources are needed forthe reader to understand the material but still at the beginning of each appen-dices there will be a part devoted to mentioning other sources which gives moredetailed exposition of the subjects covered in the respective appendices.

A Metric spaces

Metric spaces are spaces for which a metric (function which gives a distance forevery pair of points) is dened. The concept of metric spaces is important in thisthesis both from a topological point of view because the metric gives a topologyand hence the open sets of the metric spaces and because this thesis will mostlyfocus on length spaces which are metric spaces with some restrictions. Most ofthe material in this appendix can be found in either of [Mor05] or [Rud76].

Denition A.0.8 (Real coordinate space). The real coordinate space, Rn is theset of all n-tuples of R, for any positive integer n.

Example A.0.3 (Real coordinate space). An example of an element x ∈ R3 isx = [0, 1

2 , π]

A metric is a function which gives a distance (non-negative number) for everypair of elements in the set.

Denition A.0.9 (Metric). A metric on a set X is a function d : X ×X → Rthat satises for all x, y, z ∈ X the following:

a) d(x, y) = 0⇐⇒ x = y

b) d(x, y) = d(y, x) (Symmetric)

c) d(x, y) ≤ d(x, z) + d(z, y) (Triangle inequality)

Remark A.0.3. Often in the literature a fourth condition is required for a func-tion to be a metric, d(x, y) ≥ 0. This denition is equivalent with the one givenbelow due to the other three statements imply the additional condition whichis shown in Proposition A.0.1.

Example A.0.4 (Metric). The discrete metric:

d(x, y) =

0, if x = y

1, if x 6= y(A.0.21)

satises a)− c) on R. Here a)− c) follows trivially from the Denition A.0.9 ofmetric and the denition of the discrete metric.

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A METRIC SPACES

Remark A.0.4. The discrete metric will be used extensively throughout thisthesis in dierent examples because using this metric often give non-intuitiveexamples which let the reader get a better picture of what the specic object inthe example could look like.

Example A.0.5 (Metric). The function:

d(x, y) = max(x, y) (A.0.22)

on R+ is not a metric. Here b) and c) follows trivially from the Denition A.0.9of metric but a) is not fullled.

Example A.0.6 (Metric). The function:

d(x, y) = min(x, y) (A.0.23)

on R is not a metric. Here a) and c) are not fullled. For example let x = 1and y = 0. Then d(x, y) = 0 and x 6= y. Therefore a) is not fullled. Letx = 1, y = 1 and z = 0. Then d(x, y) = 1 and

d(x, z) + d(z, y) = 0, (A.0.24)

so the triangle inequality is not fullled.

Proposition A.0.1. A metric d on a set X satises the followingfor all x, y ∈ X:

d(x, y) ≥ 0. (A.0.25)

Proof. Consider:

2d(x, y) = d(x, y) + d(x, y) =

= d(x, y) + d(y, x) ≥ (by the triangle inequality)

≥ d(x, x) = 0

Because of this:d(x, y) ≥ 0. (A.0.26)

A metric space is simply a space with a metric dened on the set. Almost everyspace considered in this thesis will be metric spaces and hence these are veryimportant.

Denition A.0.10 (Metric space). A metric space is an ordered pair (X, d)where X is a set and d is metric on X.

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A METRIC SPACES

Example A.0.7 (Metric space). The ordered pair (R, d), where d is the discretemetric is a metric space.

Denition A.0.11 (Ball). Given a metric space (X, d). The open ball centeredat p ∈ X with radius r and metric d:

Bd(p, r) := x ∈ X : d(x, p) < r . (A.0.27)

A closed ball centered at p ∈ X with radius r and metric d:

Bd(p, r) := x ∈ X : d(x, p) ≤ r . (A.0.28)

Example A.0.8 (Ball). Given the metric space (Z, d) where d is the discretemetric from Example A.0.4. The open ball Bd(0, 1) equals the set 0 and theclosed ball Bd(0, 1) equals the set Z.

A set in a metric space is bounded if there is a ball which contains the set. InChapter 2 the concept of boundedly compact will be important and as the namesuggest, such sets are among other things bounded.

Denition A.0.12 (Bounded). Given a metric space (X, d). A subset Y of Xis bounded if for some x ∈ X there exists a ball Bd(x, r) of nite radius whichcontains Y . The space (X, d) is called a bounded metric space if X is bounded.

Example A.0.9 (Bounded). Given the metric space (R, d), where

d(x, y) = |x− y|. (A.0.29)

For example the intervals I[−10π,100], I(−3,3) and I[e, 72 ] are contained by Bd(0, 400),so these sets are bounded. The interval I(0,∞) is not bounded.

Example A.0.10 (Bounded). Given the metric space (R, d), where d is thediscrete metric. This space is bounded which can be seen by for exampleBd(x, r) ⊂ R for every x ∈ R where r > 1.

In a metric space, the metric induces the open sets. A set is open if there is forevery point a ball centered at the point contained in the set.

Denition A.0.13 (Open set in a metric space). Given a metric space (X, d).A subset U of X is open if for every x ∈ U , there exists an open ball Bd withcenter at x such that B ⊂ U .

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A METRIC SPACES

Example A.0.11 (Open set). Given the metric space (R, d),where

d(x, y) = |x− y|. (A.0.30)

Choose the subset

U = x ∈ R : d(0, x) < 1 . (A.0.31)

This subset U is an open set in the metric space (R, d).

Denition A.0.14 (Sequence). Given a set X. A sequence is an ordered listof elements x0, x1, x2, . . . , xk ∈ X which will be denoted xk. A sequence iscalled nite if the number of elements in the sequence is nite otherwise it isinnite.

Example A.0.12 (Sequence). Given the set R. The following list:

x0 = 1, x1 =1

2, . . . , xk =

1

k + 1, . . . (A.0.32)

is a sequence.

Remark A.0.5. Henceforth a sequence will be assumed to be innite if not oth-erwise stated.

The notion of Cauchy sequences is a corner stone in analysis and they are usedextensively both in general and in this thesis. Informally a Cauchy sequence isa sequence for which the distance between elements becomes arbitrarily smallfor later elements of the sequence. Formally it is dened as:

Denition A.0.15 (Cauchy sequence). Given a metric space (X, d). A se-quence x1, x2, x3, . . . ∈ X is a Cauchy sequence if for every ε > 0 there exists anN ∈ N such that for every n,m > N ,

d(xn, xm) ≤ ε. (A.0.33)

Example A.0.13 (Cauchy sequence). Given the metric space (R, d), where

d(x, y) = |x− y|. (A.0.34)

The sequencex0 = 0, x1 = 0.9, x2 = 0.99, x3 = 0.999, . . . (A.0.35)

is a Cauchy sequence.

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A METRIC SPACES

Example A.0.14 (Cauchy sequence). Given the metric space (R, d), where dis the discrete metric. The sequence

x0 = 0, x1 = 0.9, x2 = 0.99, x3 = 0.999, . . . (A.0.36)

is not a Cauchy sequence. This can be seen by choosing an 0 < ε < 1. Thenthere is not an N ∈ N such that whenever n,m > N and n 6= m,

d(xn, xm) ≤ ε. (A.0.37)

This is because d(xn, xm) = 1 and hence xk is not a Cauchy sequence.

When Cauchy sequences are dened, complete spaces has the dening propertythat every Cauchy sequence converges to a limit in the space.

Denition A.0.16 (Complete). Given a metric space (X, d). The space X iscomplete if every Cauchy sequence has a limit which belongs to X.

Example A.0.15 (Complete). The metric space (Q, d), where d(x, y) = |x−y|is not a complete space. For example the Cauchy sequence

3, 3.1, 3.14, 3.141, 3.1415. . . . (A.0.38)

has the limit π but π /∈ Q.

Remark A.0.6. If all the accumulation points of Q are added to Q, this new setwill be the completion of Q and in this case it is R.

Example A.0.16 (Complete). The metric space (R, d), where d(x, y) = |x− y|is a complete space.

Denition A.0.17 (Lipschitz continuous). Given two metric spaces (X, dx)and (Y, dy). A function f : X → Y is Lipschitz continuous if there exists a realC ≥ 0 such that for all x1, x2 ∈ X:

dx(f(x1), f(x2)) ≤ C · dy(x1, x2) (A.0.39)

Remark A.0.7. When a function is Lipschitz continuous it can intuitively bethought of as when the absolute value of slope of the straight line joining thefunction values is bounded by a real number. This can be seen by dividingA.0.39 by dy(x1, x2) for x1, x2 ∈ X, y1, y2 ∈ Y and this gives:

dx(f(x1), f(x2))

dy(x1, x2)≤ C. (A.0.40)

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Remark A.0.8. A real C satisfying Denition A.0.17 is a Lipschitz constant off , a minimal constant is called the dilation of f , while a map with constant 1is called non-expanding .

One classical case where Lipschitz continuity is used is as a condition in thePicard-Lindelöf theorem, which states when a solution to an ordinary dieren-tial equation exists and is unique for an initial-value problem (see [NSS11]). Inthis thesis Lipschitz continuous will be useful together with the following the-orem and Remark A.0.9 which says that if a function is shown to be Lipschitzcontinuous then it is also continuous.

Theorem A.0.4. Given metric spaces (X, dx) and (Y, dy) with a Lipschitz con-tinuous function f : X → Y . Then f is uniformly continuous on X.

Remark A.0.9. Using that uniform continuity implies continuity gives that The-orem A.0.4 implies continuity.

Proof. The proof of this theorem is omitted but can be found in [MVJ10].

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B TOPOLOGICAL SPACES AND TOPOLOGY

B Topological spaces and topology

The topic of topology deals essentially with subjects such as continuity, opensets, connectedness and compactness. These terms are generally known for stu-dents taking a couple of calculus courses but in topology they are generalizedfrom subsets of R to arbitrary sets. This thesis rely heavily on topological resultand in particular Chapter 2 uses the results from this appendix extensively. Inparticular, compactness will be very important for this thesis when studyingwhen shortest paths exist for dierent length spaces. Perhaps the most im-portant result proven in this appendix is the equivalence of what is known assequential compactness and compactness for metric spaces. This result will lateron be applied many times in several dierent proofs. Most of the results in thisappendix can be found in the text books [Rud76] and [Mor05] where especiallythe former give a thorough exposition of the basics of this subject.

B.1 Topological spaces

A topological space is a space which characterizes the topological propertiessuch as continuity, open sets, connectedness and compactness. If a space is nottopological, then crucial concepts such as continuity, open sets, etc. are notdened which are not optimal due these concepts are corner stones in analysis.Because of that the spaces in this thesis are topological. Furthermore in thissection several important topological concepts will be dened with correspond-ing examples given.

Denition B.1.1 (Closed). A set X is closed under an operation (•) if for allx, y ∈ X, x • y ∈ X.

Example B.1.1 (Closed). Given the set Z. This set is closed under the oper-ations addition (+), subtraction (−) and multiplication (·).

Example B.1.2 (Closed). Given the set Z. This set is not closed under theoperation division (/). For example for 3, 4 ∈ Z, but 3

4 /∈ Z.

Denition B.1.2 (Topological space). The set X together with a collection ofsubsets of X, denoted σ, is a topological space if:

a) ∅ and X belongs to σ

b) σ is closed under union

c) σ is closed under nite intersection

Example B.1.3 (Topological space). Given the set:

X = −3,−2,−1, 0, 1, 2, 3 (B.1.1)

and the collection of subsets σ = ∅, X. Then (X,σ) is a topological space.

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B.1 Topological spaces B TOPOLOGICAL SPACES AND TOPOLOGY

Example B.1.4 (Topological space). Given the set X = −1, 0, 1 and collec-tion of subsets σ = ∅, X, 1. Then (X,σ) is also a topological space.

Given a topological space, the topology is then the collection σ of subsets.

Denition B.1.3 (Topology). A topology to a set X is a collection of subsetsσ satisfying Denition B.1.2.

Example B.1.5 (Topology). Given the set X = −1, 0, 1. The collection ofsubsets:

σ = ∅, X, −1 , 0 , −1, 0 (B.1.2)

is a topology.

Example B.1.6 (Topology). Given the set X = −1, 0, 1. The collection ofsubsets:

σ = ∅, X, −1 , 0 (B.1.3)

is not a topology on the set X since the subset −1, 0 (which is the union−1 ∪ 0) does not belong to σ.

Now two dierent topologies will be dened. The discrete topology is denablefor every set and the standard topology is denable for every metric spaces.

Denition B.1.4 (Discrete topology). Given a set X. The discrete topology isthe collection of every subset of X.

Example B.1.7 (Discrete topology). Given the set X = −1, 0, 1. The col-lection of subsets

σ = ∅, X, −1 , 0 , 1 , −1, 0 , −1, 1 , 0, 1 (B.1.4)

is the discrete topology of X.

Now the open set of the topological spaces are the members of the topology.

Denition B.1.5 (Open set in topological spaces). Given a topological spaceX with topology σ. The open sets are the members of σ.

Example B.1.8 (Open set). Given the topology:

σ = ∅, −1, 0, 1, , −1 , 0 , −1, 0 (B.1.5)

of the topological space (X = −1, 0, 1, , σ). Then for example −1, 0 is anopen set in X.

80


The usual topology considered in metric spaces is the standard topology (natu-ral topology) which uses the metric to dene the open set in the following way:

Denition B.1.6 (Standard topology). Given a topological space (X,σ) witha metric d. Then σ is the standard topology if the following holds:

A subset of X is an open set ⇐⇒ Around every point in the set there exists anopen ball contained in the set

Example B.1.9 (Standard topology). Considering the topological space (R, σ)with the standard topology induced by the Euclidean metric d(x, y) = |x− y| inExample F.0.19. For example the following sets are open in (R, σ):

x ∈ R : 0 < x < 1,x ∈ R : −π < x < 4

229

If not otherwise specied a topological space with the standard topology hasthe Euclidean metric as its metric in this appendix.

The concept of connectedness in a topological space is not to be confusedwith another similar concept of path-connectedness. Shortly, a space is path-connected if there exists a path connecting every pair of points in the space.Connectedness, which will be dened below does not imply path-connectednessbut path-connectedness instead implies connectedness.

Denition B.1.7 (Connected). Given a topological space (X,σ). This spaceis disconnected if it is the union of two disjoint open sets, otherwise (X,σ) isconnected.

Example B.1.10 (Connected). Considering the topological space (R, σ) withthe standard topology. This space is connected because there are no two intervalswhich are both open and is a covering of R. In contrast, if the intervals neednot be open we can cover R by the intervals:

x ∈ R : ∞ < x ≤ 0 (B.1.6)

andx ∈ R : 0 ≤ x <∞ (B.1.7)

which is a covering of R but they are not disjoint open sets.

Example B.1.11 (Connected). Considering the topological space (Z, σ) withthe discrete topology. This space is disconnected. For example the open sets 0and Z \ 0 cover Z and are open sets because every set in the discrete topologyis open.

Generalizing continuous functions to functions between topological spaces areformally dened below. Intuitively, this denition can be understood by think-ing of points belonging to the same open set as being "close together" and thusthis denition will state something similar to the epsilon-delta denition.

81


Denition B.1.8 (Continuous in topological spaces). Given the topologicalspaces (X,σ1) and (Y, σ2). A function f : X → Y is continuous (continuousmap) if for all V ∈ σ2, the inverse image:

f−1(V ) = x ∈ X|f(x) ∈ V (B.1.8)

is a member of σ1 (see Figure 11).

Figure 11: A continuous function

Example B.1.12 (Continuous). Given the topological space (X,σ), where X =−1, 0, 1, and σ is the trivial topology σ = ∅, X. The function:

f : (X,σ)→ (X,σ), f(x) = x. (B.1.9)

is a continuous function.

For the open set X, the inverse image f−1(X) = X which is an open set so fis continuous.

Example B.1.13 (Continuous). Given the topological spaces (X,σ1), (Y, σ2),where X = −1, 0, 1, , Y = 0, 1, 2, , σ1 is the trivial topology σ = ∅, X andσ2 = ∅, 2, Y . The function:

f : (X,σ1)→ (Y, σ2), f(x) = x+ 1. (B.1.10)

is not continuous because f−1(2) = 1 /∈ σ1.

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For the open set X, the inverse image f−1(X) = −1, 0 is not an open set sof is not continuous.

Another key concept in topology are neighborhoods. A neighborhood to a pointis a set containing an open set and for which the point belongs to the open set.

Denition B.1.9 (Neighborhood). Given a topological space X and a pointx ∈ X. A neighborhood of x is a set V that contains an open set U such that:

x ∈ U ⊆ V. (B.1.11)

Further on the notation (X,σ) for a topological space will mostly be simpliedto X and the standard topology will be assumed in this appendix together withthe Euclidean metric unless otherwise stated.

Example B.1.14 (Neighborhood). Let the topological space be given by:

(X = −1, 0, 1, , σ) (B.1.12)

where σ is given in Example B.1.8. A possible neighborhood of 0 is the setV = −1, 0. In Example B.1.8 we see that V = −1, 0 is an open set andV ⊆ V . So V is a neighborhood of 0 in the topological space (X,σ).

One further important topological concept are accumulation points. Accumu-lation points are points in a set which can be "approximated" by other pointsin the set. This is formalized as followed.

Denition B.1.10 (Accumulation point). Given a topological space X and asubset Y ⊂ X. A point x ∈ X is an accumulation point (limit point) of Y ifevery open set containing x in X contains at least one point in Y which is not x.

Example B.1.15 (Accumulation point). Given the topological space R2 withthe standard topology and the subset

Y =

[x, y] ∈ R2 : x2 + y2 < 1\ 0. (B.1.13)

The set of accumulation points to Y is[x, y] ∈ R2 : x2 + y2 ≤ 1

. (B.1.14)

Example B.1.16 (Accumulation point). Given the topological space R2 withthe discrete topology and the subset

Y =

[x, y] ∈ R2 : x2 + y2 < 1\ 0. (B.1.15)

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The set of accumulation points to Y is ∅. This is because at every x ∈ Y , theopen set x do not contain any other points in Y .

When accumulation points are dened, the notion of closed set is easy to dene.

Denition B.1.11 (Closed set). A set is closed if it contains all its accumula-tion points.

Remark B.1.1. Another used denition for closed sets is that a set is closed if itis the complement of an open set. That these denitions are in fact equivalentfor the special case of the standard topology is shown in Lemma B.2.1.

Example B.1.17 (Closed set). Given the topological space R2 with the stan-dard topology and the subset

Y =

[x, y] ∈ R2 : x2 + y2 < 1\ 0. (B.1.16)

Here Y is not a closed set because as seen in Example B.1.15 the accumulationpoints

[x, y] ∈ R2 : x2 + y2 = 1∪ 0 (B.1.17)

is not contained in Y but instead the set[x, y] ∈ R2 : x2 + y2 ≤ 1

(B.1.18)

is a closed set.

Example B.1.18 (Closed set). Given the topological space R2 with the discretetopology and the subset

Y =

[x, y] ∈ R2 : x2 + y2 < 1\ 0. (B.1.19)

Here Y is a closed set because as seen in Example B.1.16 the set does not haveany accumulation points and ∅ ⊂ Y . In fact every set is both open and closedwith the discrete topology. This property is called a clopen set.

The notion of limits in topological spaces is readily generalized from the epsilon-delta denition as follows:

Denition B.1.12 (Limit). A limit of a sequence xk in a topological spaceX is an element a ∈ X which satises:

The element a ∈ X is a limit of xk ⇐⇒ For every neighborhood (see Deni-tion B.1.9) U of a there is an N ∈ N such that xn ∈ U for every n ≥ N .

The sequence xk is called convergent if a limit exists otherwise divergent.

84


Remark B.1.2. This is equivalent in R with: A limit of a sequence xk in R isan unique element a ∈ X which satises:

The element a ∈ X is a limit of xk ⇐⇒ For every ε > 0 there is an N ∈ R+

such that:|xn − a| < ε for every n > N. (B.1.20)

A sequence xk in Rn has a limit if every coordinate in xk separately has alimit.

Remark B.1.3. The limit of a sequence will be denoted by:

limk→∞

xk (B.1.21)

Example B.1.19 (Limit). Given the sequence xk in R, where xk = 1n . This

sequence has the limit 0. Choose an ε and let N = 1ε . Now for every n > N ,

|xn − 0| < ε.

Remark B.1.4. A sequence xk in Rn is monotone if it is increasing or decreas-ing in every coordinate of xk.

The following theorem known as the monotone convergence theorem gives thatfor monotone sequences, the existence of a limit and being bounded are equiv-alent.

Theorem B.1.1 (Monotone convergence Theorem). [Mor05] Let xk be amonotone sequence in Rn. The following statements are equivalent.

The sequence xk has a limit in Rn ⇐⇒ xk is bounded

Proof. Assume the sequence xk is in R and monotone increasing. Becausexk is bounded, the supremum a = sup

kxk is nite. Now choose an ε, then

there exists a xN such that:

xN > a− ε (B.1.22)

because otherwise a would not be a supremum.

Now utilizing that xk is increasing gives for all n > N ,

|xn − a| = a− xn ≤ a− xN < ε, (B.1.23)

where the last inequality is derived from (B.1.22). But this is the denitionthat a is the limit of xk. Exactly the same reasoning can be used to show

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B.2 Compactness B TOPOLOGICAL SPACES AND TOPOLOGY

that if xk is decreasing, it has a limit, a = infkxk. This gives that a bounded

sequence in R has a limit.

If xk is a sequence in Rn the same procedure as above can be used for everycoordinate of xk, to show that it has a limit.

B.2 Compactness

In this section a number of important result concerning compact spaces will begiven and to successfully accomplish this some further topological concepts areneeded. Perhaps the most important result in this section is the equivalence ofcompactness and sequential compactness (which will be dened later) in metricspaces.

The compactness of a space is another property of the topology. This notion isvery abstract and hard to be put into word for a general topological space. Ina metric space, compactness can be thought of as when choosing points fromthe set, eventually two of the point needs to be arbitrarily close together. Theabstract denition of compactness is given below.

Denition B.2.1 (Compact). Given a topological space X. The space X iscompact if for every collection of open subsets of X, Uii∈A, such that:

X ⊂⋃i∈A

Ui, (B.2.1)

there is a nite subcovering, i.e there is a nite B ⊆ A such that:

X ⊂⋃i∈B

Ui. (B.2.2)

Sometimes this denition is hard to work with and due to this the main resultin this section is the equivalence of sequential compactness and compactness inmetric spaces. In some cases in this thesis, sequential compactness will be easierto work with in some proofs, specically in Chapter 2.

Before the equivalence of the two types of compactness is proven, another resultfor subsets of En called the Heine-Borel Theorem B.2.1 will be proven. Thisstates that a subset X of En is compact if and only if X is closed and bounded.

To prove this theorem, interior points will be used. An interior point to a subsetis a point such that, there exists a neighborhood around the point contained inthe subset.

Denition B.2.2 (Interior point). Given a topological space X with metric d.A point x is an interior point of a subset Y ⊆ X if there exists an open ballBd(x, r) such that:

Bd(x, r) ⊂ Y. (B.2.3)

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Remark B.2.1. Note that with the standard topology, every point in an openset is an interior point.

Example B.2.1 (Interior point). Given the topological space R2 with the dis-crete metric. The set R2 has each of its points as an interior point which canbe seen by choosing x ∈ R2 and the open ball:

Bd(x, 1) ⊂ R2. (B.2.4)

Example B.2.2 (Interior point). Given the topological space R2 with the dis-crete metric. The set:

X =

[x, y] ∈ R2 : x2 + y2 ≤ 1

(B.2.5)

has each of its points as an interior point which can be seen by choosing x ∈ Xand the open ball:

Bd([x, y], 1) ⊂ X. (B.2.6)

Example B.2.3 (Interior point). Given the topological space R2 with the Eu-clidean metric. The set:

X =

[x, y] ∈ R2 : x2 + y2 ≤ 1

= Bd(0, 1) (B.2.7)

have points which are not interior points. More precisely, every point such that:

x2 + y2 = 1 (B.2.8)

is not an interior point of X because there does not exists an open ball contain-ing such a point and such that the open ball is contained in X.

The following three lemmas will be used to prove the Heine-Borel TheoremB.2.1. The rst of these lemmas concerns when a complement to an open set isa closed set.

Lemma B.2.1. [Rud76] Given a topological space X with the standard topol-ogy. Then the following statements are equivalent:

A subset Y ⊂ X is open ⇐⇒ The complement Y c is a closed set.

Proof. =⇒: Assume Y is open.

Choose an accumulation point y ∈ Y c. Then every open ball around y containsa point x ∈ Y c. This gives that y is not an interior point of Y and since Y isan open set, y ∈ Y c.

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⇐=: Assume Y c is closed.

Choose y ∈ Y . Then y /∈ Y c and y is not an accumulation point of Y c. Thusthere exists an open ball B around y such that:

B ∩ Y c = ∅. (B.2.9)

Then B ⊂ Y and thus y is an interior point of Y .

The following lemma is not only essential to prove the Heine-Borel theorem butwill also be used repeatedly in Chapter 2.

Lemma B.2.2. [Rud76] Given a topological space with the standard topology.Then a closed subset X of a compact set Y is compact.

Proof. Consider the collection BX of open set which covers X. Now the setU = Xc is an open set and the collection:

BY ∪ U (B.2.10)

is an open covering of Y . Because Y is compact there exists a nite subcoveringof BY ∪ U ,

B′Y ∪ U (B.2.11)

which covers Y . The sets X and U are disjoint by construction which makesthe collection:

B′X = B′Y \ U (B.2.12)

a nite subcovering of X.

The last lemma is known as the Cantor's intersection lemma. This is a technicalresult which states, a sequence of closed and bounded subsets of En which are"nested" has a non-empty intersection.

Lemma B.2.3 (Cantor's intersection lemma). [Rud76] Given a sequence Xkof closed and bounded subsets of En such that:

X0 ⊇ X1 ⊇ · · ·Xk ⊇ Xk+1 · · · (B.2.13)

This gives the following: ⋂k

Xk 6= ∅ (B.2.14)

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Proof. Consider the sequence ak, where ak is the inmum of Xk at each co-ordinate and since Xk is closed ak ∈ Xk . This sequence will be monotonicincreasing, i.e will be increasing for every coordinate separately and since it isbounded, it must have a limit a from Theorem B.1.1.

Now choose any i ≥ 0. The subsequence akk≥i is contained in Xk and con-verges to a. Since Xk is closed, a ∈ Xk. This is true for every Xk so:

a ∈⋂k

Uk (B.2.15)

Now the ground work to prove the Heine-Borel theorem is ready and what fol-lows is the theorem with a corresponding proof.

Theorem B.2.1 (Heine-Borel theorem). [Rud76] Given a subset X of En.The following statements are equivalent:

X is compact ⇐⇒ X is closed and bounded

Proof. =⇒: Assume X is compact.

First observe the following: Let a be an accumulation point in X. Then anynite collection C of open sets, where the open sets U ∈ C are such that thereexist neighborhoods VU of a. These open sets are not an open covering whichcan be seen by considering the intersection of the neighborhoods VU of a. Thisset is a neighborhoodW of a because every neighborhood contains open set andthe intersection of open sets is open, so W contains an open set and thereby isa neighborhood. Since a is an accumulation point there has to exists a b ∈ Wwhich belongs to X. From this it follows that C is not a covering.

Assume X is not closed. Then there exists an accumulation point a such thata /∈ X. Now consider the collection D of neighborhoods Ux of x for every x ∈ Xwhere Ux are open sets, chosen such that they do not intersect a neighborhoodVa of a. This collection D is an open covering of X. Now considering any nitesubcollection of D. This cannot be an open covering of X as seen above withC. But then X is not compact which is a contradiction. So X is closed.

Consider the collection E of open balls:

Bd(x, r)r∈R+ . (B.2.16)

This is an open covering of X. Now every open subcovering of X must havea largest ball which contains every other ball in that subcovering. This ballbounds X. Because of this X is bounded.

⇐=: Assume X is closed and bounded.

Since X is a bounded subset of Rn, a box

Z0 = [−a, a]n (B.2.17)

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which is given by the condition that every dimension of R lies in the intervalI[−a,a], can always enclose X. From Lemma B.2.2 it is sucient to show thatZ is compact for X to be compact.

Assume that Z0 is not compact. Now there exists an open covering F of Z0

such that there is no nite subcovering of F. Construct 2n boxes by bisectingthe sides of Z0. These boxes will now have the diameter a.

Because Z0 has not a nite subcovering there exists a box Z1 with an innitecovering from F of the 2n boxes constructed. Now Z1 can be bisected to 2n newboxes. Repeating this produces a sequence of boxes:

Z0 ⊇ Z1 ⊇ · · ·Zk ⊇ Zk+1 · · · (B.2.18)

The side length 2a2k

of the boxes will tend to 0 as k goes to innity. Since thesequence Zk is closed and bounded the Lemma B.2.3 gives that there existsa x such that:

x ∈⋂k

Zk. (B.2.19)

Since F covers Z0, there exists an U ∈ F such that x ∈ F. Because U is open,there exists a ball Bd(x, r) such that:

Bd(x, r) ∈ U. (B.2.20)

Choosing large enough k, there is a Zk such that:

Zk ⊇ Bd(x, r) ⊇ U. (B.2.21)

But this gives that the innite subcovering from F of Zk can be replaced by U ,which yields a contradiction, so Z0 is compact.

Remark B.2.2. Now the Heine-Borel Theorem B.2.1 gives an easy result forgiving examples of compact sets in En.

Example B.2.4 (Compact). The sets:

X =

[x, y] ∈ R2 : x2 + y2 ≤ 1

(B.2.22)

andY =

[x, y, z] ∈ R3 : 3x2 − 5y2 + 2z2 ≤ 5

(B.2.23)

are compact in E2 respectively E3.

Example B.2.5 (Compact). The set Rn is not compact with the standard topol-ogy. This is because the set is not bounded.

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Remark B.2.3. Another often used denition of compactness is that a set X iscompact if every sequence in X has a subsequence which converge to a pointin X. This type of compactness is not equivalent with Denition B.2.1 andis sometimes called sequentially compact . But they are equivalent in En (andmore generally in metric spaces as seen further on in this appendix) and theequivalent of the Heine-Borel Theorem B.2.1 for this type of compactness iscalled the Bolzano-Weierstrass theorem.

Theorem B.2.2 (Bolzano-Weierstrass theorem). [Mor05] Every bounded se-quence in Rn has a convergent subsequence.

Proof. First assume that the sequence xk is real and nonnegative. If xk hassome negative terms, translate it to a new sequence yk such that every termis nonnegative which is possible due to xk is bounded and prove that thissequence has a convergent subsequence. Then translate back the convergentsubsequence in yk to a convergent subsequence in xk.

Because the sequence is nonnegative and bounded there exists an integer part(all integers before the comma) C before the comma sign which occur innitelymany times in the sequence. Let the rst element in the subsequence be xm1

=C. Now only consider the innitely many elements with the integer part C.Then there exists a rst decimal place c1 which occurs innitely many times inthe new sequence. Let the second element in the subsequence be xm2

= C.c1Continue this process to construct a subsequence:

xmk = C.c1c2 . . . ck . . . . (B.2.24)

The subsequence xmk converges to

a = C.c1c2 . . . (B.2.25)

due to choosing ε > 0. Then there trivially exists an integer N , such that forevery integer i > N ,

d(xmi − a) < ε. (B.2.26)

Because xmk will have at least the rst N decimals in common with a.

If xk is a sequence in Rn, construct by the above scheme for every coordinateof xk separately, convergent subsequences:

xm1k

,xm2k

, . . . ,

xmnk

. (B.2.27)

Then the sequence Xk, where

Xk = [xm1k, . . . , xmnk ] (B.2.28)

is a convergent subsequence.

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Before being able to prove the equivalence of compactness and sequential com-pactness in metric spaces several new concepts needs to be introduced.

First of these is the concept of countability. It deals with the cardinality of setsand more specically it gives a way to categorize sets with innite cardinalityas countable or uncountable.

Denition B.2.3 (Countable). A countable set X is a set with nite cardinal-ity or a set for which there exists a bijection E.0.11 f : N → X between thenatural numbers and X, otherwise X is uncountable.

The following proposition gives a useful method to prove countability of a setby a diagonalization process.

Proposition B.2.1. [Mor05]The set Q is countable.

Proof. The set Q is clearly not of nite cardinality so there has to be a bijectionbetween the natural numbers N and Q for Q to be countable. It is sucientto show for the positive rational numbers that such a bijection exists becauseotherwise just alternate between positive and negative number in this fashionfor a given list of positive rational numbers ai:

1↔ a0, 2↔ −a0, 3↔ a1, 4↔ −a1, . . . . (B.2.29)

Arrange the positive rationals as in Figure 12.

Start the listing at the upper left corner with 1/1 and move through the diag-onals as intended in the Figure 12 and skip new numbers in the list which hasalready been repeated like, 1/1 = 2/2. This procedure will produce a bijectionof the form:

1↔ 1/1, 2↔ 2/1, 3↔ 1/2, 4↔ 3/1, 5↔ 1/3, 6↔ 4/1, 7↔ 3/2, 8↔ 2/3, 9↔ 1/4, . . . .(B.2.30)

Remark B.2.4. The method used above in the proof of the countability of Q iscalled the Cantor's diagonalization argument .

Example B.2.6 (Countable). The sets Z, 2Z, N, X = 0, 1, 2, 3, 4 are count-able.

This proposition shows that R is not countable and hence the cardinality of Z,Q is not equal to R.

Proposition B.2.2. [Mor05]The set R is uncountable

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Figure 12: A counting of the rationals

Remark B.2.5. Hence on a listing of a countable set will be a bijection as theseseen in the proof of Proposition B.2.1.

Proof. Assume R is countable. Then the positive reals need to be countable.Take a listing of these. Construct an α such that it diers in the rst decimalplace from the rst decimal for the rst element in the list, such that it diersin the second decimal place from the second decimal for the second element inthe list and so on. This α will not be listed and we have a contradiction. Thenthe set R is uncountable.

The next needed concept is denseness. Subsets are dense if every point in the setis an accumulation point. Below is a formalization of the concept of denseness.

Denition B.2.4 (Dense). A subset Y of a topological space X is dense if forevery x ∈ X, x belongs to Y or is an accumulation point of Y .

Example B.2.7 (Dense). The subset Q of R is dense in R. To show that thisis the case, choose x ∈ R such that x /∈ Q. But there is not an open set whichcontains x but do not contain any points in Q. Therefore x is an accumulationpoint of Q and Q is dense.

Example B.2.8 (Dense). The subset of irrational numbers, R \Q is dense inR. This shows that a set can have more then one dense subset and in the earlierexample the cardinality of Q is lower then of R but the cardinality of R\Q equalsthat of R.

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Example B.2.9 (Dense). The subset Q of R is not dense in the topologicalspace R with the discrete topology. This is easily realized when considering thatevery subset of R is open. Choose x ∈ R such that x /∈ Q. The open set x donot contain any points in Q, so x is not an accumulation point and therefore Qis not dense in R with the discrete topology.

The next concept is separability. A set is separable if it has a countable densesubset.

Denition B.2.5 (Separable). A topological space X is separable if it containsa countable dense subset.

Example B.2.10 (Separable). Given the space R. The subset Q is a countabledense subset of R and hence R is separable.

The last topological property needed is the notion of a base for a topologicalspace. This concept is formalized below.

Denition B.2.6 (Base). A collection Uα of open sets in a topological spaceX is a base for X if for every x ∈ X and every open set U ⊂ X with x ∈ U ,there is an α such that:

x ∈ Uα ⊂ U. (B.2.31)

Example B.2.11 (Base). Given the topological space R with the standard topol-ogy. The collection of open sets formed by the topology of R is a base for R.

Example B.2.12 (Base). Given the topological space R with the discrete topol-ogy. The collection of open sets formed by the topology of R is not a base for R.To see this choose x ∈ R and the open set x. Now there is clearly not an openset U such that x ∈ U ⊂ x.

The proof of the equivalence between sequential compactness and compactnessB.2.3 in metric spaces is divided into the four following lemmas and then themain theorem.

The rst Lemma B.2.4 gives that every separable metric space has a countablebase.

Lemma B.2.4. [Rud76] Given a separable metric space X and the standardtopology. Then there exists a countable base for X

Proof. Using the denition of a separable space there is a countable dense subset:

C = c1, c2, . . . (B.2.32)

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Figure 13: An element of a base

of X. Consider the countable collection of open sets:

B = Bd(ci, r) : r ∈ Q, i = 1, 2, . . . . (B.2.33)

What remains to show is that B is a base for X.

Choose x ∈ X and an open set U such that x ∈ U . Then from the denitionof open set in the standard topology B.1.6 there is an open ball Bd(x, r) suchthat:

Bd(x, r) ⊂ U, (B.2.34)

without loss of generality assume that r ∈ Q. From the denition of dense, x isan accumulation point of C and hence the ball:

Bd(x, r/2) (B.2.35)

contains at least one element of C. This says that there is an i such that

d(x, ci) < r/2. (B.2.36)

This gives that the ball:Bd(ci, r/2) ∈ B (B.2.37)

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satises:x ∈ Bd(ci, r/2) ⊂ U (B.2.38)

and hence C is a base.

The next Lemma B.2.5 says that a sequential compact space is separable.

Lemma B.2.5. [Rud76] Given a sequentially compact space X. Then X isseparable.

Proof. Fix δ > 0 and x1 ∈ X. Now choose x2 ∈ X such that:

d(x1, x2) ≥ δ. (B.2.39)

Continue choosing x1, x2, . . . , xj+1 such that:

d(xi, xj+1) ≥ δ for i = 1, 2, . . . , j. (B.2.40)

This gives a sequence with a distance of at least δ between each element. Thiscan not be done innitely because otherwise there would not exists a convergingsubsequence of xj and this would give a contradiction because X is sequen-tially compact. Now choose j such that for every point in X, the distance tosome xi is less than δ. This is possible by choosing xj+1 ∈ X, which gives that:

d(xj+1, xi) < δ (B.2.41)

for some i = 1, 2, . . . , j. Continue choosing xj+2, xj+3, . . . , xj+h until:

d(xk, xl) < δ (B.2.42)

for every pair xk, xl. This can be done by choosing xj+i in the following way: Let

r = maximin d(xi, xj) : for j = 1, 2, . . . , j + 1 and j 6= i . (B.2.43)

Then take the xi with the minimum index such that:

min d(xi, xj) : for j = 1, 2, . . . , j + 1 and j 6= i = r (B.2.44)

and denote as xk. Finally choose xj+i ∈ Bd(xk, r). Now the nite set xii=1,2,...,j+h

satises that for every point in X, the distance to some xi is less than δ. Thisgives that: ⋃

j=1,2...,j+h

Bd(xj , δ) = X. (B.2.45)

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Let δ = 1n for n = 1, 2, . . .. For n = 1 this yields the sequence

x1j

and n = 2

x 12 j

. The set:

B =x 1n i

: n = 1, 2, . . . , i = 1, 2, . . . , j

(B.2.46)

is countable. Choosing x ∈ X and a ball Bd(x, r). This ball contains a y ∈ Xsuch that y 6= x because for 1

n < r there exists a x 1n i

such that:

d(y, x 1n i

) < r, (B.2.47)

hence B is dense in X and thus X is separable.

Remark B.2.6. The two Lemmas B.2.4, B.2.5 gives together that every sequen-tially compact space X has a countable base.

The next Lemma B.2.6 is of a technical nature and states that every open cov-ering of a space with countable base has a countable subcovering.

Lemma B.2.6. [Rud76] Given a space X with countable base. Then everyopen covering of X has a countable subcovering.

Remark B.2.7. The type of compactness in Lemma B.2.6 that every open cov-ering of a space X has a countable subcovering is often called Lindelöf compact-ness.

Proof. Let Vj be a base for X and choose an open covering Uα of X. Nowlet J be the set of all j such that there exists an α for which Vj ⊂ Uα.

The set Vjj∈J is an open covering of X which is seen by: choose x ∈ X, thenthere exists an α such that x ∈ Uα. From the denition of a base there exists aVi such that:

x ∈ Vi ⊂ Uα. (B.2.48)

Then i ∈ J andx ∈

⋃j∈J

Vj . (B.2.49)

Furthermore for each j ∈ J, choose αj such that Vj ⊂ Uαj . This gives that:⋃j∈J

Vj ⊆⋃j∈J

Uαj = X (B.2.50)

and henceUαj

is a countable subcovering.

The last Lemma B.2.7 gives that the intersection of a sequence of sets is non-empty if every set is a subset of the previous sets in the sequence.

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Lemma B.2.7. [Rud76] Given a sequence of non-empty closed subsets Xnof a sequentially compact set X such that X1 ⊃ X2 ⊃ . . .. Then:

∞⋂i=1

Xi 6= ∅. (B.2.51)

Proof. Choose xn ∈ Xn for each n and consider the set:

A = xn : n = 1, 2, . . . . (B.2.52)

If A is nite then there exists a xk which belongs to innitely many Xn. BecauseX1 ⊃ X2 ⊃ . . . this gives that xk belongs to the intersection and hence is notempty.

Thus assume that A is innite. Because X is sequentially compact, the sequenceA has a converging subsequence with a limit a ∈ X and hence is an accumula-tion point. Now for a x value of n, every neighborhood U around a containsinnitely many points of A and among these there is such that xi ∈ U for i ≥ n.This gives that:

xi ∈ Xi ⊂ Xn. (B.2.53)

Since every neighborhood of a contains a point of Xn, a is an accumulationpoint of Xn and Xn is closed gives that a ∈ Xn. This reasoning applies forevery n and thus

a ∈∞⋂i=1

Xi. (B.2.54)

Now nally Theorem B.2.3 is ready to be proven. This theorem is repeatedlyused in Chapter 2.

Theorem B.2.3. [Rud76] Given a metric space X. Then the following state-ments is fullled:

X is a compact space ⇐⇒ X is a sequentially compact space

Proof. =⇒: Assume X is a compact space.

Assume that X is not sequentially compact. Now it is possible to choose a se-quence S = xk ∈ X without a subsequence converging in X. Then S cannothave an accumulation in X. This gives that there is an open covering Ux∀x∈Xof S such that each Ux has at most one point in common with S. But theredoes not exists a nite subcovering of Ux∀x∈X which covers S, is not compactand therefore X cannot be compact which gives a contradiction. Thus X issequentially compact.

⇐=: Assume X is a sequentially compact space.

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From Lemmas B.2.4, B.2.5 gives that X has a countable base. Choose an opencovering Uα and by Lemma B.2.6, X is Lindelöf compact, i.e. every opencovering has a countable subcovering. Let Ui be such a covering. If Ui isnite the proof is complete, hence assume Ui is innite and thus

U1 ∪ U2 ∪ . . . ∪ Un + X (B.2.55)

for every n.

Now let

Xn = x ∈ X : x /∈ U1 ∪ U2 ∪ . . . ∪ Un = X ∩ U c1 ∩ U c2 ∩ . . . U cn. (B.2.56)

BecauseU1 ∪ U2 ∪ . . . ∪ Un (B.2.57)

is open, the setXn is closed due to Lemma B.2.1 and furthermoreX1 ⊃ X2 ⊃ . . .so using Lemma B.2.7 gives that:

∞⋂i=1

Xi 6= ∅ (B.2.58)

and thus there exists a y such that:

y /∈∞⋃i=1

Ui (B.2.59)

which is a contradiction because Ui is a covering of X. This gives that Uiis a nite subcovering a X is compact.

The next Theorem B.2.9 is needed in the Lebesgue´s lemma but before provingthis two lemmas will be stated and proved.

The rst lemma gives that the image of a continuous function on a compact setis compact.

Lemma B.2.8. [Mor05] Given compact set X and a continuous function f .Then the image of f is a compact set.

Proof. Choose an open covering Uα of f [X]. Because f is continuous, f−1[Uα]is an open set. Considering the union:⋃

α

f−1[Uα] = f−1

[⋃α

Uα

]= f−1[f [X]] = X (B.2.60)

and thus f−1[Uα]

(B.2.61)

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is an open covering of X. Because X is compact there exists a nite subcoveringof (B.2.61), Uni=1. Rewriting f [X] gives:

f [X] = f [

n⋃i=1

f−1[Ui]] = f

[f−1[

n⋃i=1

Ui]

]⊆

n⋃i=1

Ui (B.2.62)

which gives that Uni=1 is a nite subcovering of f [X] and thus f [X] is compact.

This Lemma B.2.9 states that a real function on a compact set has a maximumand a minimum.

Lemma B.2.9. [Mor05] Given a compact set X and a continuous functionf : X → R. Then f has a maximum and a minimum.

Proof. From Lemma B.2.8 it follows that f [X] is compact and using Heine-BorelTheorem B.2.1 gives that f [X] is closed and bounded. Let a = sup f [X] andb = inf f [X]. Because f [X] is closed, a and b belongs to f [X] and are thusmaximum and minimum of f [X] respectively.

Remark B.2.8. A continuous function f : X → R on a compact set X has amaximum if there exists an element f(x0) such that:

f(xo) ≥ f(x) for every x ∈ X (B.2.63)

and a minimum if there exists an element f(x0) such that:

f(xo) ≤ f(x) for every x ∈ X. (B.2.64)

The Lebesgue´s Lemma B.2.4 gives that there is always possible to nd a ballwith positive radius which is contained in an open set belonging to an opencovering. This theorem is of a technical nature and is used in Chapter 2.

Theorem B.2.4 (Lebesgue´s Lemma). [BBI01] Given a compact metric spaceX. Let Uα be an open covering of X. Then there exists a ρ > 0 such thatany ball of radius ρ is contained in some of the sets Uα.

Remark B.2.9. The number ρ in Theorem B.2.4 is often called Lebesgue numberof the covering.

Proof. The metric of X is nite because otherwise it is not possible to create anopen covering. Further assume that none of the Uα covers the whole X becauseotherwise the proof is completed. Then create the function f : X → R, where

f(x) = sup r ∈ R : Bd(x, r) is contained in some Uα . (B.2.65)

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This function is well-dened because Uα is an open covering and f is positivefor every x ∈ X and it is continuous because f is Lipschitz continuous, whichis seen by noting that

d(f(x1), f(x2)) ≤ d(x1, x2) (B.2.66)

and thus continuous by Theorem A.0.9. Then using Lemma B.2.9 gives that fhas a minimum a. Now let ρ = a/2.

Curves plays an important part of the theory in this thesis and of especial im-portance in Chapter 2. The reader should check the dierence between curvesand paths (see Denition 2.1.1).

Denition B.2.7 (Curve). Given a topological space X. A curve γ is a con-tinuous map γ : I→ X, where I is an interval.

Example B.2.13 (Curve). Given the topological space R2 with the standardtopology. The continuous map:

γ(t) =

x = t, t ∈ I[0,3]

y = t2, t ∈ I[0,3]

, (B.2.67)

where I[0,3] is the closed interval between 0 and 3, and thus γ(t) is a curve in R2.

Example B.2.14 (Curve). Given the topological space R with the discrete topol-ogy. The map:

γ(t) =

1, t ∈ Q0, t ∈ R \Q

(B.2.68)

is continuous so γ is a curve.

In Chapter 2 shortest paths and geodesics will play a central part but to be ableto dene these there is needed some kind of method to measure the length of acurve and this is done in the following way.

Denition B.2.8 (Length of curve). Given a topological space X with metricd. The length of a curve γ : I[a, b]→ X with a partition:

a = t1 < t2 . . . < tn = b (B.2.69)

is given by:


n−1∑i=1

d(γ(ti), γ(ti+1)) (B.2.70)

Here the supremum is taken over all possible partitions of I[a, b] and n is un-bounded. The curve length l is called rectiable if l is nite.

101


Remark B.2.10. Two curves γ1 and γ2 with equal images γ1(I[a, b]) and γ2(I[c, d])have the same length.

Almost all spaces in this thesis will be what is called Hausdor spaces. A Haus-dor space in a metric space is essentially a space in which there is a positivedistance between every distinct points in the space and this is generalized in thefollowing matter.

Denition B.2.9 (Hausdor space). A topological space is a Hausdor spaceX if every two distinct points are separable by neighborhoods, i.e. for givenpoints x and y there exist neighborhoods U ,V to x, y respectively such that:

U ∩ V = ∅. (B.2.71)

Example B.2.15 (Hausdor space). Given the topological space (Z, σ) whereσ is the discrete topology. Then any members a, b ∈ Z is separable by the neigh-borhoods U = a and V = b respectively.

Example B.2.16 (Hausdor space). Given the topological space (R, σ) whereσ is the discrete topology. Then any members a, b ∈ R is separable by the neigh-borhoods U = a and V = b respectively.

Example B.2.17 (Hausdor space). Given the topological space (R, σ) whereσ is the trivial topology σ = ∅,R. This space is not a Hausdor space whichcan been seen by none of the members are separable by neighborhoods of the opensets ∅,R. For example choose x, y ∈ R. Neither x nor y belongs to ∅, so x, ymost be separated by neighborhoods U, V , where R ⊆ U and R ⊆ V . But this isnot true due to U ∩ V 6= ∅.

Denition B.2.10 (Domain). Given a nite dimensional vector space X. Adomain Ω is any connected open subset of X.

Example B.2.18 (Domain). Given the vector space R. The set

x ∈ R : 0 < x < 1 (B.2.72)

is an open set in R and connected, so this set is a domain in R.

Example B.2.19 (Domain). Given the vector space R. The set

x ∈ R : 0 < x < 1 ∩ x ∈ R : 2 < x < 3 (B.2.73)

is an open set in R but it is not connected, so this set is not a domain in R.

102


Example B.2.20 (Domain). Given the vector space R2. The set:

X = Bd(0, r) \R+ (B.2.74)

is an open set in R2.

A convex set in Rn is set which satises that every points can be joined by astraight line segment.

Denition B.2.11 (Convex set). Given the space Rn. A subset X ⊂ Rn isa convex set if every pair of points (x, y) ∈ X can be joined by a straight linesegment γ such that γ ∈ X.

Figure 14: A non convex set

Example B.2.21 (Convex set). Every ball in the Euclidean space En is a con-vex space but no member of the set of sets constructed by taking the set of ballsin En and removing the center point are a convex space. This can be seen bythe example of the unit ball B centered at [0, 0] in E2 with [0, 0] removed andchoosing the points x = [0.5, 0], y = [−0.5, 0]. Then the line segment between xand y is not in B because the line segment contains [0, 0].

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C ALGEBRA

C Algebra

The eld of algebra deals with operations and relations between sets. The rstexample of such operations are addition and multiplication of numbers whichare taught in elementary school. Generalizing this concepts gives rise to the eldof abstract algebra where the objects groups, rings and elds are the conceptsof main focus. In this thesis the theory surrounding elds will be used covertlyin that sense that a lot of maps will be mapped into elds and thus a basicunderstanding of elds is useful for understanding the material in mainly Chap-ter 3 and 4. For those readers wanting a more thorough examination of thiseld a basic textbook in abstract algebra as for example [Fra67] is recommended.

An equivalence relation is a relation on a set which relates if two elements inthe set are "equivalent" or not. In order to dene this strictly, the concepts ofCartesian products and binary relations are needed.

Denition C.0.12 (Cartesian product). The Cartesian product between twosets X and Y is dened by:

X × Y = (x, y)|x ∈ X, y ∈ Y . (C.0.75)

Example C.0.22 (Cartesian product). Given the sets X = 0, 1 andY = 1, 2. The Cartesian product of X and Y is given by:

X × Y = [0, 1], [0, 2], [1, 1], [1, 2] . (C.0.76)

Denition C.0.13 (Binary relation). A binary relation R is an ordered triplet(X,Y,G), where X and Y are sets and G is a subset of the Cartesian productX × Y . Here X is the relations domain, Y the codomain and G the graph.

Remark C.0.11. A binary relation (X,X,G) is a binary operator if G satises:

G : X ×X → X. (C.0.77)

This property makes a binary operator closed.

Example C.0.23 (Binary relation). One binary relation R = (X,Y,G) is givenby:

R = (X = P, Y = N, G = [2, 0], [2, 2], [2, 4], ...[3, 0], [3, 3], [3, 6], ...), (C.0.78)

where P is the set of primes, N is the set of natural numbers and G is constructedby a member p ∈ P is associated with every member of N which is a multiple of p.

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C ALGEBRA

Denition C.0.14 (Equivalence relation). Given graph ∼ on a set X (in Def-inition C.0.13 we have the following triplet (X,X,∼)). ∼ is an equivalencerelation if the following is satised for all x, y, z ∈ X:

a) x ∼ x (Reexive)

b) x ∼ y =⇒ y ∼ x (Symmetric)

c) x ∼ y and y ∼ z =⇒ x ∼ z (Transitive)

Remark C.0.12. Two elements a and b satises a ∼ b if the element [a, b] of theCartesian product belongs to the graph ∼.

Example C.0.24 (Equivalence relation). One example of an equivalence rela-tion on Z is equal to (a = b). Using the terminology in Denition C.0.14 wehave the triplet (Z,Z, G), where

G = [0, 0], [1, 1], [−1,−1], [2, 2], [−2,−2], ... . (C.0.79)

Example C.0.25 (Equivalence relation). Another example of an equivalencerelation on Z is modulus (a = b mod(n)). The triplet is now (Z,Z, G) for n = 3,where

G = [0, 0], [0, 3], [0,−3], [0, 6], [0,−6], ..., [1, 1], [1, 4], [1,−2], [1, 7], [1,−5], ... .(C.0.80)

Example C.0.26 (Equivalence relation). One example of a relation which isnot an equivalence relation on Z is greater than (≤). It satises the reexivityand the transitivity but it is not symmetric. For example 0 ≤ 1 but this does notimply 1 ≤ 0.

An equivalence class to a specic element is then the natural extension of lettingevery equivalent element to the specic element belong to the same equivalenceclass.

Denition C.0.15 (Equivalence class). Given a set X and an equivalencerelation ∼. The equivalence class of x0 ∈ X is the set:

[x0]X = x ∈ X : x0 ∼ x . (C.0.81)

Example C.0.27 (Equivalence class). The equivalence class of 1 for modulus3 for the set N is:

[1]N = 1, 4, 7, 10, ... . (C.0.82)

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C ALGEBRA

In abstract algebra, the rst object often considered is a group. A group is aset together with an operation dened on this set. This operation shall satisfythe conditions of closure, associativity, existence of identity and inverse element.The meaning of these conditions will be claried in the following denition butone reason for this abstract denition is to get a generalization of the propertiesof for example addition on the set of integers.

Denition C.0.16 (Group). A group is a set G and an operation (•) thatsatises for all a, b, c ∈ G:

a) a • b ∈ G (Closed)

b) (a • b) • c = a • (b • c) (Associative)

c) There exists an element e ∈ G such that:

e • a = a • e = a (Identity element) (C.0.83)

d) To each a there exists an element a−1 ∈ G such that:

a • a−1 = a−1 • a = e (Inverse element) (C.0.84)

Example C.0.28 (Group). The set Z under addition (+) is a group. Here theidentity element is given by 0 and for example the inverse element of an elementa ∈ Z is given by −a.

Example C.0.29 (Group). The set Z under multiplication (·) is not a group.The identity element exists and is given by 1 but the inverse element only existsfor the element 1.

Example C.0.30 (Group). The set X = 0, 1 under addition and modulo 2is a group. Here the identity element is 0 and the inverse element of 1 is 1 and0 is 0.

When a group is dened, an abelian group is simple a group which is commu-tative.

Denition C.0.17 (Abelian group). A group (G, •) is abelian (commutative)if for all a, b ∈ G:

a • b = b • a. (C.0.85)

Example C.0.31 (Abelian group). The group in Example C.0.28 is an Abeliangroup due to the addition operator being abelian, i.e. a+ b = b+ a.

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C ALGEBRA

Example C.0.32 (Abelian group). The set Z under the operation subtraction(−) is not an abelian group. This is due the operator not being abelian, forexample 0− 1 6= 1− 0 and not associative, (1− 2)− 3 = −4 and 1− (2− 3) = 2.

A eld is an extension of a group by which instead of only dening one operationon the set, two operations are dened and not only shall the set together withon of the operations be a group (not strictly true, see Denition C.0.18) butalso the two operations need to satisfy the distributive law. It is strictly denedin the following way.

Denition C.0.18 (Field). A eld is a set F together with two operations(+), (•) (denote the operations addition and multiplication) that satises for alla, b, c ∈ F :

a) (F,+) is an abelian group

b) (F \ 0 , •), is an abelian group where 0 is the additive identity element

c) a • (b+ c) = (a • b) + (a • c) (Distributive)

Example C.0.33 (Field). The set R with the two operations additive addition(+) and multiplication (·) is a eld. (R,+) is an abelian group with identityelement 0 and the inverse element of a ∈ R is −a. The set (R \ 0 , ·) is alsoan abelian group with identity element 1 and inverse element to a is 1

a . Thedistributivity for + and · is also fullled because:

a · (b+ c) = (a · b) + (a · c). (C.0.86)

Example C.0.34 (Field). The set of rational numbers Q with the two opera-tions addition (+) and multiplication (·) is a eld. (Q,+) is an abelian groupwith identity element 0 and the inverse element of a

b ∈ Q is −ab . The set(Q \ 0 , ·) is also an abelian group with identity element 1 and inverse elementto a

b is ba .The distributivity for + and · is also fullled because:

a

b· ( cd

+e

f) = (

a

b· cd

) + (a

b· ef

). (C.0.87)

Example C.0.35 (Field). The set Z with the two operations addition (+) andmultiplication (·) is not a eld. (Z,+) is an abelian group with identity element0 and the inverse element of a ∈ Q is −a. But neither (Z, ·) nor (Z \ 0 , ·) isan abelian group due to the inverse element does not exists except for 1 ∈ Z.

When a eld is dened, a subeld is a subset which contains the two identityelements and also is a eld.

Denition C.0.19 (Subeld). A subeld F1 of a eld F is a subset that con-tains the additive identity element 0 and the multiplicative identity elementdenoted as 1, and such that F1 is also a eld.

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C.1 Linear algebra C ALGEBRA

Example C.0.36 (Subeld). The eld (Q,+, ·) is a subeld of the eld (R,+, ·).

Remark C.0.13. Observe that Z is not a subeld of R or Q.

Denition C.0.20 (Permutation). Given a set X. A permutation σ is a bijec-tion (see Denition E.0.11) σ : X → X.

Example C.0.37 (Permutation). Given the set X = 1, 2, 3, 4. A permuta-tion σ of X is given by:

σ =

(1 2 3 42 1 4 3

)(C.0.88)

where the above notation means that 1 is mapped on 2, 2 is mapped on 1, 3 ismapped on 4 and 4 is mapped on 3.

C.1 Linear algebra

The goal of this section is to give the theory necessary for Sylvester's criterionwhich is a condition when a matrix is positive-denite. This will be needed inChapter 4. For a more complete exposition of this subject see [HJ85].

The determinant is a value associated with an n × n-matrix. Several dierent(and equivalent) denitions exist of determinants and one of these is DenitionC.1.1 stated below. Note that the matrix specied in the denition is not neces-sarily real-valued nor complex-valued but instead only need to belong to a eldssuch that the operations + and · are specied.

Denition C.1.1 (Determinant). Given an n× n matrix A. Let Σ be the setof all permutations σ of 1, 2, . . . , n. Then the determinant of A, det(A) isgiven by:

det(A) =∑σ∈Σ

sgn(σ)

n∏i=1

ai,σ(i) (C.1.1)

where sgn(σ) is dened as 1 if σ does an odd number of reorderings of 1, 2, . . . , nand −1 if σ does an even number of reorderings, for example

σ =

(1 2 3 42 1 4 3

), (C.1.2)

σ does an even number (2) of reorderings so sgn(σ) = −1. The entity ai,σ(i)

is the element of the matrix A which corresponds to the i:th row and σ(i):thcolumn.

109


Example C.1.1 (Determinant). Given a 2 × 2 matrix A =

[a bc d

]. The de-

terminant of A is then given by:

∑σ∈Σ

sgn(σ)2∏i=1

ai,σ(i) = sgn([1, 2])2∏i=1

ai,σ(i) + sgn([2, 1])2∏i=1

ai,σ(i) =

= a1,1a2,2 − a1,2a2,1 = ad− bc.

A complex matrix is dened as positive-denite if the following is satised:

Denition C.1.2 (Positive-denite). Given a matrix A where each elementaij ∈ C. The matrix A is positive-denite if

zTAz > 0 (C.1.3)

for every non-zero vector z ∈ C.

Example C.1.2 (Positive-denite). Given the matrix A =

[0 ab 0

], where

a, b ∈ R+. Then A is positive-denite matrix because choosing z = [z1, z2] 6= 0gives:

zTAz = 0 + z2z2b+ z1z1a+ 0 = Re(z22)b+Re(z2

1)a > 0. (C.1.4)

Example C.1.3 (Positive-denite). Given the matrix

A =

2 −1 0−1 2 −10 −1 2

(C.1.5)

and the vector z = [z1, z2.z3]. This gives for zj = xj + iyj:

zTAz =[(2z1 − z2) (−z1 + 2z2 − z3) (−z2 + 2z3)

] z1

z2

z3

=

= 2z1z1 − z2z1 − z1z2 + 2z2z2 − z3z2 − z2z3 + 2z3z3 =

= 2Re(z21)− 2(x1x2 + y1y2) + 2Re(z2

2)− 2(x2x3 + y2y3) + 2Re(z23) =

= Re(z21) +Re(z2

3) + (x1 − x2)2 + (y1 − y2)2 + (x2 − x3)2 + (y2 − y3)2

and thus zTAz > 0 if z 6= 0. This example shows that matrices with negativevalued elements can be positive-denite.

Example C.1.4 (Positive-denite). Given the matrix A =

[1 32 1

]and the

vector z = [1,−1], gives that:zTAz = −3 (C.1.6)

110


This shows that it is not a sucient condition for a matrix to have positiveelements for it to be positive-denite.

A complex matrix is an Hermitian matrix if it equal to its conjugate transpose.

Denition C.1.3 (Hermitian matrix). A matrix A is an Hermitian matrix iffor each element aij of A:

aij = aji. (C.1.7)

Example C.1.5 (Hermitian matrix). The matrix:[π 3 + 2i

3− 2i 1721

](C.1.8)

is an Hermitian matrix.

The Sylvester`s criterion gives the equivalence of positive deniteness and thatevery upper left corner submatrix has a positive determinant. This gives insome cases an easy method of checking whether a matrix is positive-denite ornot.

Theorem C.1.1 (Sylvesters criterion). Given an Hermitian n × n matrix A.Then the following conditions are equivalent:

a) A is positive-denite

b) The following matrices has positive determinants:

The upper left 1-by-1 corner of AThe upper left 2-by-2 corner of A

...The upper left n-by-n corner of A

Proof. The proof of this theorem is omitted here but can be found in [Gil91]

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D VECTOR SPACES

D Vector spaces

In this Appendix, vector spaces together with some concepts often associatedwith vector spaces such as inner product, norms, basis, dual spaces, tensors anddierential forms will be dened and be given examples of. Most of the resultsand concepts in this appendix will be used in Chapter 3 and Chapter 4. To thereader who wish to study these subjects closer the books [AR10] and [Car06]are recommended.

A vector space is a eld whose elements are called vectors and satisfying the eightconditions in Denition D.0.4. One important vector space which is studied inthis thesis is the tangent space Denition 3.1.14. The additive and multiplica-tive properties among others makes vector spaces easy to work with.

Denition D.0.4 (Vector space). A vector space (linear space) over a eld Fis a set X with two binary operators +, · where (+) operates between elementsof X and (·) between an element of F and X. For X to be a vector space thefollowing is satised for every u, v, w ∈ X and a, b ∈ F

a) u+ (v + w) = (u+ v) + w (Associative)

b) u+ v = v + u (Commutative)

c) There exists an element 0 ∈ X called the zero vector such that:

v + 0 = 0 + v = v (Additive identity element) (D.0.9)

d) There exists an additive inverse −v ∈ X such that:

v + (−v) = −v + v = 0 (Inverse element) (D.0.10)

e) a · (u+ v) = (a · u) + (a · v) (Distributive)

f) (a+ b) · v = (a · v) + (b · v) (Distributive)

g) a · (b · v) = (a · b) · v (Associative)

h) There exists an element 1 ∈ F such that 1 · v = v (Multiplicative identity)

Remark D.0.1. The condition that +, · are binary operators insures that:

u+ v ∈ X and a · u ∈ X. (D.0.11)

Example D.0.6 (Vector space). The set R with the operations addition (+)and multiplication (·) over the eld R is a vector space. For example the zerovector is 0, the multiplicative identity element is 1 and the additive identity el-ement is 0.

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D VECTOR SPACES

A linear map is a function, mapping elements of a vector space to another vectorspace such that the function is both additive and homogeneous.

Denition D.0.5 (Linear map). Given two vector spaces X1 and X2 over aeld F . A linear map L is a function L : X1 → X2 that satises for all x, y ∈ X1

and c ∈ F :

a) L(x+ y) = L(x) + L(y) (Additive)

b) L(cx) = c · L(x) (Homogeneity)

Example D.0.7 (Linear map). Given the vector space R with the operationsaddition (+) and multiplication (·) over the eld R. Then the map L : R→ R,L(x) = cx is a linear map. This can be seen for all x, y, c, d ∈ R by:

a) L(x+ y) = c(x+ y) = cx+ cy = L(x) + f(y)

b) L(dx) = dcx = d · L(x)

A basis for a vector space is a subset of the vector space which is linearly inde-pendent and spans the whole vector space and by which means:

Denition D.0.6 (Basis). A basis E = v1, ..., vn of a vector space X over aeld F is a nite subset of X which satises for all

a1, ..., an ∈ K and v1, ..., vn ∈ V : (D.0.12)

a) If a1v1 + ...+ anvn = 0 =⇒ a1, ..., an = 0 (Linearly independent)

b) For every x ∈ X it is possible to choose a1, ..., an ∈ K such that:

x = a1v1 + ...+ anvn. (Spanning) (D.0.13)

Example D.0.8 (Basis). Consider the vector space R2 (see Denition A.0.8)with the operations addition (+) and multiplication (·) over the eld R2. Thevectors e1 = [1, 0] and e2 = [0, 1] form a basis for R2. This basis is called thestandard basis. Other vectors x, y satisfying being basis for R2 are for examplex = [1, 2] and y = [2, 1].

Example D.0.9 (Basis). Given the vector space in Example D.0.8. The vec-tors x = [1, 2], y = [2, 4] are not a basis for R2. It is easily visible that x, y arelinearly dependent by: 2x− y = 0.

For every given vector space a corresponding dual space is dened as the set ofevery linear map from the vector space to the corresponding eld. In this thesismany objects in Chapter 3 and Chapter 4 will belong to dual spaces so dualspaces will be an important concept.

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D VECTOR SPACES

Denition D.0.7 (Dual space). Given a vector space X over a eld F . Thedual space X∗ to X is given by the set of all linear maps (linear functionals)f : X → F .

Example D.0.10 (Dual space). The dual space of the vector space (R,+, ·)over the eld R is the maps:

f : f(x) = cx, c, x ∈ R . (D.0.14)

That in fact all linear maps in Example D.0.10 are of the form f(x) = cx canbe seen by the following reasoning:

Let g be a linear map g : R → R of the vector space (R,+, ·) over the eld R.Then:

g(x) = g(x · 1) = x · g(1) (D.0.15)

is of the form:f(x) = cx (D.0.16)

due to the fact that g(1) ∈ R.

Then the dual basis is simply dened as:

Denition D.0.8 (Dual basis). Given a vector space W . A dual basis is abasis for X∗.

Example D.0.11 (Dual basis). The basis for the dual space in Example D.0.10is any member of R\ 0.

Many geometrical objects are tensors. For example scalars, vectors and lin-ear maps are examples of dierent tensors. Two of the rst to study whatis today known as tensors where Ricci and Levi-Civita during the late 19:thcentury when developing the theory surrounding the curvature of Riemannianmanifolds. There are several dierent equivalent denitions of tensors used indierent books and the one used in this thesis denes a tensor as a multilinearmapping satisfying the following condition:

Denition D.0.9 (Tensor). Given a vector space X over a eld F . An (n×m)tensor is a map:

T : X∗ ×X∗ × . . .×X∗︸︷︷︸n times

×X ×X × . . .×X︸︷︷︸m times

→ F (D.0.17)

which is linear in every argument (multi linear).

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D VECTOR SPACES

Example D.0.12 (Tensor). Given the vector space Rn and

x = [x1, . . . , xn] ∈ Rn. (D.0.18)

Then the mean function:

E : Rn → R, E(x) =

n∑i=1

xin

(D.0.19)

is a tensor.

To be able to talk about length of vectors and orthogonality in vector spaces aninner product needs to be dened. The inner product is a mapping which takesa pair of vectors and maps them on the scalars.

Denition D.0.10 (Inner product). Given a subeld F of the complex numbersC and a vector space X. An inner product 〈·, ·〉 is a map:

〈·, ·〉 : X ×X → F (D.0.20)

which satises for all x, y, z ∈ X and a ∈ F :

a) 〈x, y〉 = 〈y, x〉 (Conjugate symmetric)

b) 〈ax, y〉 = a〈x, y〉 and 〈x+ y, z〉 = 〈x, z〉+ 〈y, z〉 (Linear)

c) 〈x, x〉 ≥ 0 and if 〈x, x〉 = 0 =⇒ x = 0 (Positive-denite)

Example D.0.13 (Inner product). Consider the vector space R over the eldR with the operations addition (+) and multiplication (·). The map:

〈x, y〉 = xy (D.0.21)

is an inner product which is easily visible by:

a) 〈x, y〉 = xy = yx = 〈y, x〉 = 〈y, x〉.

b) 〈ax, y〉 = axy = a〈x, y〉 and〈x+ y, z〉 = (x+ y)z = xz + yz = 〈x, z〉+ 〈y, z〉

c) 〈x, x〉 = x2 ≥ 0 and if 〈x, x〉 = x2 = 0 =⇒ x = 0

Example D.0.14 (Inner product). The inner product in Denition F.0.12 forx, y ∈ X, where x = [x1, ..., xn] and y = [y1, ..., yn] are given by:

〈x, y〉 = x · y =

n∑i=1

xiyi (D.0.22)

This is a generalization of the inner product in Example D.0.13 for n-dimensions:

a) 〈x, y〉 = x · y = y · x = 〈y, x〉 = 〈y, x〉.

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D VECTOR SPACES

b) 〈ax, y〉 =n∑i=1

axiyi = an∑i=1

xiyi = a〈x, y〉 and

〈x+ y, z〉 =n∑i=1

(xi + yi)zi =

=n∑i=1

xizi +n∑i=1

yizi = 〈x, z〉+ 〈y, z〉

c) 〈x, x〉 =n∑i=1

x2i ≥ 0 and if

〈x, x〉 =n∑i=1

x2i = 0 =⇒ x = 0

Example D.0.15 (Inner product). Consider the vector space R over the eldR with the operations addition (+) and multiplication (·). The map:

< x, y >= x+ y (D.0.23)

is not an inner product because:

a′) 〈x, y〉 = x+ y = y + x = 〈y, x〉 = 〈y, x〉

b′) 〈ax, y〉 = (ax) + y 6= a〈x, y〉 and〈x+ y, z〉 = (x+ y) + z 6= 〈x, z〉+ 〈y, z〉

c′) 〈x, x〉 = 2x 0 and if 〈x, x〉 = 2x = 0 =⇒ x = 0

So 〈x, y〉 = x+ y satises a) but not b) nor c) in Denition D.0.10.

Naturally, an inner product space is then a vector space with a dened innerproduct.

Denition D.0.11 (Inner product space). An inner product space X is a vec-tor space over the subeld F of C with an inner product 〈., .〉.

Example D.0.16 (Inner product space). The vector space R over the eld Rwith the inner product for x, y ∈ R, 〈x, y〉 = xy is an inner product space.

The following theorem proves a useful inequality named the Cauchy-Schwarzinequality. It is useful in many branches in mathematics and probability theoryand will be used a couple of times in proofs and examples in this thesis.

Theorem D.0.2 (Cauchy-Schwarz inequality). Given the inner product spaceX and elements x, y ∈ X.

The following is satised: |〈x, y〉|2 ≤ 〈x, x〉 · 〈y, y〉 and is called the Cauchy-Schwarz inequality.

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D VECTOR SPACES

Proof. Given the inner product space X, elements x, y ∈ X and t ∈ R. Assume〈x, y〉 ∈ R.

Consider: 〈tx+ y, tx+ y〉 = 〈tx, tx+ y〉+ 〈y, tx+ y〉 =

= 〈tx+ y, tx〉+ 〈tx+ y, y〉 =

= t2〈x, x〉+ t〈x, y〉+ t · 〈x, y〉+ 〈y, y〉 (by using t ∈ R)

= t2〈x, x〉+ 2t ·Re(〈x, y〉) + 〈y, y〉 (by using 〈x, y〉 ∈ R)

= t2〈x, x〉+ 2t · 〈x, y〉+ 〈y, y〉

This is a quadratic equation in t of type

at2 + bt+ c (D.0.24)

wherea = 〈x, x〉, b = 2 · 〈x, y〉 (D.0.25)

and c = 〈y, y〉. Furthermore from the denition of inner product D.0.10:

〈tx+ y, tx+ y〉 ≥ 0, (D.0.26)

so:

at2 + bt+ c ≥ 0 and a, c ≥ 0, b ∈ R (D.0.27)

Assume a 6= 0 and considering (D.0.27) yields:

at2 + bt+ c ≥ 0 ⇐⇒ t2 + ba t+ c

a ≥ 0 ⇐⇒

⇐⇒ (t+ b2a )2 ≥ − c

a + b2

4a2 = b2−4ac4a2

Setting t = − b2a which minimize (t+ b

2a )2 yields:

b2 − 4ac ≤ 0 (D.0.28)

So applying (D.0.28) yields:

4|〈x, y〉|2 − 4〈x, x〉〈y, y〉 ≤ 0⇐⇒ |〈x, y〉|2 ≤ 〈x, x〉 · 〈y, y〉 (D.0.29)

If a = 0, the equation (D.0.27) is

bt+ c ≥ 0 (D.0.30)

which implies, b = 0 is the only solution for all t due to c ≥ 0. This solutionalso satises (D.0.28), so the solution of (D.0.27) is (D.0.28).

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D VECTOR SPACES

Assume〈x, y〉 ∈ C. Choose λ such that |λ| = 1 and

〈x, λy〉 = |〈x, y〉| (D.0.31)

Considering: |〈x, y〉|2 = 〈x, λy〉2 = (Using 〈x, λy〉 ∈ R)

= |〈x, λy〉|2 ≤ (Using Cauchy-Schwarz inequality for real valued inner products)

≤ 〈x, x〉〈λy, λy〉 = 〈x, x〉 · (λλ)〈y, y〉 = 〈x, x〉〈y, y〉

In vector spaces, norms are functions which assign lengths to vectors, that is anon-negative number for each vector and only zero for the zero vector. Furtherconditions for a function to be a norm on a vector space is given below.

Denition D.0.12 (Norm). Given a vector space X over a subeld F of C. Anorm is a function ‖·‖ : V → F such that for all x, y ∈ V and a ∈ F :

a) ‖ax‖ = |a| · ‖x‖ (Positive homogeneous)

b) ‖x+ y‖ ≤ ‖x‖+ ‖y‖ (Triangle inequality)

c) If ‖x‖ = 0 =⇒ x = 0 (Zero vector)

The following Theorem D.0.3 gives an easy method for creating a norm from aninner product.

Theorem D.0.3. Given an inner product space X over the eld F with innerproduct 〈·, ·〉. Then the following is a norm for x, y ∈ X and a ∈ F :

‖x‖ =√〈x, x〉 (D.0.32)

Proof. For ‖x‖ =√〈x, x〉 to be a norm it needs to satisfy a)-c) in Denition

D.0.12.

a) ‖ax‖ =√〈ax, ax〉 =

√aa〈x, x〉 = |a|

√〈x, x〉

b) To show the triangle inequality consider:

‖x+ y‖2 = 〈x+ y, x+ y〉 = 〈x, x+ y〉+ 〈y, x+ y〉 =

= ‖x‖2 + 〈x, y〉+ 〈y, x〉+ ‖y‖2 ≤

≤ ‖x‖2 + 2|〈x, y〉|+ ‖y‖2 ≤ (using the Cauchy-Schwarz inequality)

≤ ‖x‖2 + 2‖x‖ · ‖y‖+ ‖y‖2 = (‖x‖+ ‖y‖)2

Taking the square rot of this expression shows b).

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D.1 Dierential forms D VECTOR SPACES

c) Using the denition of inner product D.0.10 and square rot mapping 0 on 0proves c).

Remark D.0.2. In an inner product space with inner product 〈·, ·〉, the normwhich will always be used is: ‖x‖ =

√〈x, x〉.

Example D.0.17 (Norm). Given the vector space Rn in Denition A.0.8 overthe eld R. The norm dened in Denition F.0.12 for x ∈ Rn, ‖x‖ =

√x · x is

a norm given by the inner product D.0.14 satisfying Theorem D.0.3.

Example D.0.18 (Norm). Given the vector space R over the eld R. Thefunction f(x) = x, x ∈ R is not a norm for x, y, a ∈ R.

a′) f(ax) = ax = a · f(x)

b′) f(x+ y) = x+ y = f(x) + f(y)

c′) If f(x) = x = 0 =⇒ x = 0

So f satises b) and c) in Denition D.0.12 but not a).

D.1 Dierential forms

Dierential forms is a subject studied, among others, the elds dierential ge-ometry and tensor analysis. Élie Cartan is usually counted as one of the pioneersin this subject due him publishing the paper "Sur certaines expressions diéren-tielles et le problème de Pfa" [Car99] in 1899 giving a more formal treatment ofdierential forms. Perhaps the rst dierential forms one encounter and a goodexample are expressions of the form f(x)dx where f(x) is an integrand. Thelayout and material in this section of the appendix follow close that of [Car06]so for those readers interested in a more in depth study of dierential forms thisbook is a good start.

Before dening dierential forms a couple of others concepts needs to be dened.After dening dierential forms, the operation exterior product will be denedfor dierential for together with some rules of calculation.

The rst needed concept is alternating forms. One possible denition of alter-nating forms is given below:

Denition D.1.1 (Alternating multilinear form). Given vector spacesX1, . . . , Xn

over a eld F . A multilinear form is a map:

f : X1 × . . .×Xn → F (D.1.1)

which satises, for each i, the map:

f(x1, . . . , xi, . . . , xn) (D.1.2)

120


is a linear map of xi if all but the variable xi is constant and a map with nvariables which is a multilinear form is an n-linear form.

If the spaces X1, . . . , Xn are equal, the map f is an alternating map if

f(x1, . . . , xn) = 0 (D.1.3)

whenever xi = xi+1 for some 1 ≤ i < n.

Example D.1.1 (Alternating multilinear form). Given an n × n matrix A.The determinant det(A) is an alternating multilinear form. This can be seenby letting every column ci of A correspond to the vector space X over a eld F .Then the map:

X × . . .×X → F (D.1.4)

given by the determinant is a multilinear map and an alternating map whichcan be seen from the dening equation of a determinant (see Denition C.1.1).

det(A) =∑σ∈Σ

sgn(σ)

n∏i=1

ai,σ(i) (D.1.5)

The following proposition is a useful result for alternating form which also couldhave served as a denition.

Proposition D.1.1. Given an alternating form f(x1, x2).Then:

f(x1, x2) = −f(x2, x1). (D.1.6)

Proof. Using the linearity of f and that f is alternating gives:

f(x1, x2) = f(x1, x2)+f(x1, x1) = f(x1, x2+x1) = (add and substract f(x2, x2+x1))

= f(x1 + x2, x2 + x1)− f(x2, x2 + x1) =

= −f(x2, x2 + x1) = −f(x2, x1)

Remark D.1.1. Using the property in Proposition D.1.1 gives that an equivalentdenition of an alternating form is that f(x1, . . . , xn) = 0 whenever x1, . . . , xnis linearly dependent.

The next goal is to dene the operation exterior product on alternating forms.Before doing this the following function class, n-linear continuous alternatingmappings are needed.

121


Denition D.1.2 (n-linear continuous alternating mappings). The space of n-linear continuous alternating mappings Xn → Y where X,Y are normed vectorspaces is denoted by An(X,Y ).

Let f ∈ An1(X,Y ), g ∈ An2

(X,Z) together with a bilinear continuous map:

φ : Y × Z → V, (D.1.7)

where X,Y, Z, V are normed vector spaces. Now create the mappingh : Xn1+n2 → V by:

h(x1, . . . , xn1+n2) = φ(f(x1, . . . , xn1), g(xn1+1, . . . , xn1+n2)) (D.1.8)

Clearly h is multilinear and continuous but not generally alternating. Insteadit is only alternating when considered as a function of the rst n1 variables orof the last n2. Denote the space of such maps An1,n2

(X,V ).

Now dene an association between members h ∈ An1,n2(X,V ) and h′ ∈ An1+n2

(X,V )by:

ϕn1,n2: An1,n2

(X,V )→ An1+n2(X,V ), (D.1.9)

where ϕn1,n2(h) is the multilinear mapping:

h′ =∑σ∈S

sgn(σ)h(xσ(1), . . . , xσ(n1+n2)), (D.1.10)

where the sum is over the set S of all permutations σ of 1, . . . , n1 + n2 satis-fying:

σ(1) < . . . < σ(n1), (D.1.11)

σ(n1 + 1) < . . . < σ(n1 + n2). (D.1.12)

This mapping is in fact an alternating form which is shown in the followingproposition:

Proposition D.1.2. [Car06] The mapping h′ dened in (D.1.10) is an alter-nating map.

Proof. Divide the permutations satisfying (D.1.11) into two cases and let xi =xi+1:

a) Both σ−1(i) and σ−1(i + 1) are less than or equal to n1 or that both arelarger or equal to n1 + 1. From equation (D.1.10) and that h is alternatingwhen considering the rst n1 variables or the last n2 variables, h′ is 0.

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b) The case left is that σ−1(i) ≤ n1 and σ−1(i+1) ≥ n2 or that σ−1(i+1) ≤ n1

and σ−1(i) ≥ n2. Consider the permutation τ given by replacing i and i+ 1.

Now using τ , pair each permutation satisfying σ−1(i) ≤ n1 andσ−1(i+ 1) ≥ n2 with a permutation satisfying σ−1(i+ 1) ≤ n1 andσ−1(i) ≥ n2. This gives rise to pairs which satises when added together:

sgn(σ)h(xσ(1), . . . , xσ(n1+n2))− sgn(σ)h(xτ(σ(1)), . . . , xτ(σ(n1+n2))) = 0.(D.1.13)

The expression (D.1.13) is zero for every pair because xi = xi+1, thus everyterm in the sum (D.1.10) is zero and hence h′ is alternating.

An exterior product is a mapping between two multilinear continuous alternat-ing mappings with respect to a continuous bilinear mapping in the following way:

Denition D.1.3 (Exterior product). Given alternating multilinear formsf ∈ An1

(X,Y ), g ∈ An2(X,Y ) in the vector spaces Vf , Vg respectively. The

exterior product ∧ over f, g relative to a bilinear continuous map:

φ : Vf × Vg → V, (D.1.14)

where V is a vector space is dened as the map:

ϕn1,n2(h) ∈ An1+n2

(X,V ) (D.1.15)

where h ∈ An1,n2(X,V ) is dened as the element h = (f ∧φg) satisfying:

(f ∧φg)(x1, . . . , xn1+n2

) =∑σ∈S

sgn(σ) φ(f(xσ(1), . . . , xσ(n1)), g(xσ(n1+1), . . . , xσ(n1+n2))),

(D.1.16)

where S is the set of permutations satisfying (D.1.11).

Remark D.1.2. The ∧ will be omitted in the notation f ∧φg when Vf = V ,

Vg = R andφ : Vf × Vg → V (D.1.17)

is simply multiplication between a vector and a scalar.

Example D.1.2 (Exterior product). Let n1 = 1 in the denition of the exteriorproduct. Then the exterior product (f ∧

φg)(x1, . . . , xn2+1) is given by:

(f ∧φg)(x1, . . . , xn2+1) =

n2+1∑i=1

(−1)i+1φ(f(xi), g(x1, . . . , xi−1, xi+1, . . . , xn2+1)).

(D.1.18)

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Now it is time to dene dierential forms. It is a mapping from an open set tothe space of n-linear continuous alternating mappings.

Denition D.1.4 (Dierential form). Given normed vector spaces X,Y andan open set U ∈ X. A mapping:

f : U → An(X,Y ) (D.1.19)

is a dierential n-form dened in U with values in Y and f is said to be of classCk if it is k-times continuously dierentiable.

Remark D.1.3. Using the notation in the denition of a dierential form, a0-form is dened as a mapping f : U → Y and also that the space of n-formsdened in U with values at Y which is of class Ck and is denoted as Ω

(k)n (U, Y ),

is a vector space.

Example D.1.3 (Dierential form). Given a function f : U → R, f(x, y, z)which is smooth and U is an open set U ∈ R3. The map:

df =∂f

∂xdx+

∂f

∂ydy +

∂f

∂zdz (D.1.20)

is a 1-form dened in U with values in A1(R3,R) of class C∞ and is thus inthe space Ω

(∞)1 (U,R).

When dierential forms have been dened, operations on these objects can bespecied. The exterior product for multilinear alternating forms have alreadybeen dened and following the same approach for dierential forms, the exteriorproduct for dierential forms comes naturally.

Given vector spaces X1, X2, X3 and a continuous linear map:

φ : X1 ×X2 → X3. (D.1.21)

Now consider elements:

a ∈ Ωkn1(U,X1), b ∈ Ωkn2

(U,X2) (D.1.22)

where U is an open set in the vector space V . For x ∈ U , the elements a(x) andb(x) are in the spaces An1(V,X1), An2(V,X2) respectively. Then the exteriorproduct between these elements is given by:

a(x)∧φb(x) ∈ An1+n2

(V,X3) (D.1.23)

Now the mapping:

µ : U → An1+n2(V,X3), (D.1.24)

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whereµ(x) = a(x)∧

φb(x), (D.1.25)

is of class Ck due to a ∈ Ωkn1(U,X1) and b ∈ Ωkn2

(U,X2).

Denition D.1.5 (Exterior product for dierential forms). Given the dieren-tial forms a, b, normed vector spaces X1, X2, X3, V and an open set U ∈ V .The exterior product of a and b relative to the map

φ : X1 ×X2 → X3 (D.1.26)

is the dierential form:

µ : U → An1+n2(V,X3), (D.1.27)

where:µ(x) = a(x)∧

φb(x). (D.1.28)

Remark D.1.4. If the dierential forms a and b are of class Cn, the dierentialform µ in the denition is of class Cn. Furthermore if a and b are n1, n2-formsrespectively the dierential form µ is a n1 + n2-form.

a∧φb(x; ξ1, . . . , ξn1+n2

) is explicitly given by:

a∧φb(x; ξ1, . . . , ξn1+n2) =

∑σ∈S

sgn(σ) φ(a(x; ξσ(1), . . . , ξσ(n1)), b(x; ξσ(n1+1), . . . , ξσ(n1+n2))),

(D.1.29)

using (D.1.16).

Example D.1.4 (Exterior product for dierential forms). Given a function

f : U → X2, where U ⊂ X1 is an open subset, X1, X2, X3 are normed vector

spaces and the n-form µ : U → An(X3,R). Let the map φ be dened by vector

multiplication between X2 and R. Then the exterior product f ∧φµ is given by:

f ∧φµ(x; ξ1, . . . , ξn) =

∑σ∈S

sgn(σ) φ(f(x), µ(x; ξ1, . . . , ξn) =

= f(x) · µ(x; ξ1, . . . , ξn),

where f · µ is the vector product.

Example D.1.5 (Exterior product for dierential forms). Given dierential1-forms a, b with scalar values and the map φ : R × R → R which is scalar

125


multiplication. Then the exterior product a ∧ b(x; ξ1, ξ2) is given by:

a ∧ b(x; ξ1, ξ2) = a(x; ξ1)b(x; ξ2)− a(x; ξ2)b(x; ξ1) (D.1.30)

The next goal is to associate an operation d on µ ∈ Ωkn(U,X) such that

dµ ∈ Ωk−1n+1(U,X). (D.1.31)

Let the map:µ : U → An(X1, X) (D.1.32)

be of class Cn, where U ⊂ X1 is an open set. Now considered the dierentiationof the map µ, which will be a map:

µ′ : U → A1,n(X1, X) (D.1.33)

that satises for each x ∈ X1 and i ∈ 1, . . . , n,

(µ′(x)ξi)(ξ1, . . . , ξi−1, ξi+1,...,ξn) ∈ X. (D.1.34)

This gives that µ′ is n-linear and an alternating function when considering thevariables ξ1, . . . , ξi−1, ξi+1, . . . , ξn and thus µ′(x) ∈ A1,n(X1, X).

The composite of the map (D.1.31) and φ1,n : A1,n(X1, X) → An+1(X1, X)from Denition D.1.3 is a map dµ satisfying (D.1.31).

Denition D.1.6 (Exterior dierentiation). Given normed vector spacesX1, X.The exterior dierential dµ of a dierential form µ is a map which consists of acomposition of two maps:

µ′ : U → A1,n(X1, X), (D.1.35)

φ1,n : A1,n(X1, X)→ An+1(X1, X). (D.1.36)

Explicitly dµ(x; ξ1, . . . , ξn) = φ1,n(µ′(x; ξ1, . . . , ξn)) is given by:

dµ(x; ξ1, . . . , ξn) =

n∑i=1

(−1)i+1(µ′(x) · ξi) · (ξ1, . . . , ξi−1, ξi+1, . . . , ξn). (D.1.37)

Remark D.1.5. An n-form µ of class Ck has an exterior dierential dµ which isan n+ 1-form of class Ck−1.

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Example D.1.6 (Exterior dierentiation). Given a dierential form µ ∈ Ωk1(U,X),where U ∈ X1 and X1, X are normed vector spaces. Then the exterior dieren-tial of µ is given by:

dµ(x; ξ1, ξ2) = (µ′(x) · ξ1) · ξ2 − (µ′(x) · ξ2) · ξ1. (D.1.38)

Example D.1.7 (Exterior dierentiation). Given a function f : U → X. Thenthe exterior dierential of f is given by:

df(x; ξ) = f ′(x) · ξ. (D.1.39)

The following theorems in this section will give a number of usable rules forcalculations regarding dierentials and dierential forms.

The rst theorem shows the result of taking the exterior dierential of a dier-ential form multiplied with a function. The result resembles the usual "productrule" when dierentiating the product of two functions.

Theorem D.1.1. [Car06] Given a function f of class C1 and a dierentialn-form µ. Then:

d(f · µ) = (df) ∧ µ+ f · (dµ). (D.1.40)

Proof. Let f : U → X1, µ : U → X2 where X1, X2, X3 are normed vectorspaces, U ∈ X where U is an open set in the normed vector space X and let

φ : X1 ×X2 → X3 (D.1.41)

be a continuous bilinear mapping. Let the mapping:

f · µ : U → An+1(X1, X) (D.1.42)

be given by:f · µ(x) = φ(f(x), µ(x)). (D.1.43)

Now dierentiating this function with respect to x at a vector ξ ∈ X gives:

(f · µ)′(x; ξ1, . . . , ξn) = φ(f ′(x) · ξ, µ(x)) + φ(f(x), µ′(x) · ξ). (D.1.44)

Then using (D.1.37) with µ replaced with f · µ gives:

d(f · µ)(x; ξ1, . . . , ξn) =

n∑i=1

(−1)i+1((f · µ)′(x) · ξi) · (ξ1, . . . , ξi−1, ξi+1,...,ξn)

(D.1.45)

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Combining (D.1.44) and (D.1.45) together gives:

d(f ·µ)(x; ξ1, . . . , ξn) =

n∑i=1

(−1)i+1φ(f ′(x)·ξ,µ(x))+

n∑i=1

(−1)i+1φ(f(x), µ′(x)·ξi)

(D.1.46)

and continuing with this expression (D.1.46) gives (D.1.40) as seen in [Car06].

The next theorem gives the result of taking the exterior dierential of a exteriorproduct of dierential forms.

Theorem D.1.2. [Car06] Given dierential forms a ∈ Ωkn1(U,R), b ∈ Ωkn2

(U,R).Then:

d(a ∧ b) = (da) ∧ b+ (−1)n1a ∧ (db). (D.1.47)

Proof. The proof of this theorem is omitted here but can be found in [Car06].

This theorem gives that taking the exterior dierential two times on a dieren-tial forms gives zero.

Theorem D.1.3. [Car06] Given a dierential form µ ∈ Ωkn1(U,R) where

k ≥ 2. Then:

d(dµ) = 0. (D.1.48)


The following three theorems will be representation theorems where dierentdierential forms will be represented in a "canonical" way.

Let X be a nite m-dimensional space. Now choosing a basis for X gives anidentication between X and Rm. Let ui ∈ L(Rm,R) be the i:th coordinatefunction where L(Rm) is the space of linear functions from Rm to R. Given anopen set U ⊂ Rm, let xi be the dierentiable map U → R such that xi is therestriction of ui to U .

Then the exterior dierential of xi is the constant mapping (i.e every elementin the domain is mapped on a single element) U → L(Rm,R), on the elementui ∈ L(Rm,R).

Then the rst representation theorem gives the following canonical representa-tion of a dierential form.

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Theorem D.1.4. [Car06] Given an open set U ∈ R and a dierential formµ ∈ Ωmn (U,X). Then µ can be represented uniquely in the following canonicalform:

µ =∑

i1<...<in

ci1,...,in(x)dxi1 ∧ . . . ∧ dxin , (D.1.49)

where ci1,...,in are functions U → X of class Ck and i1, . . . , in are integerssatisfying:

i1 < . . . < in, 1 ≤ i1, . . . , in ≤ m. (D.1.50)


The next representation theorem deals with how the exterior dierential of afunction can be represented canonically.

Theorem D.1.5. [Car06] Given an open set U ⊂ Rm, a normed vector spaceX and a function f : U → X of class C1. Then the exterior dierential of fcan be represented on the form:

df =

m∑i=1

∂f

∂xidxi. (D.1.51)


Let henceforth in this appendix U be an open set in Rm. The next thing toconsider is the exterior multiplication of two dierential forms canonically repre-sented. Because exterior multiplication is distributive with respect to addition,let the two dierential n1, n2 forms a,b respectively only consist of one term:

a = α(x)dxi1 ∧ . . . ∧ dxin1(D.1.52)

b = β(x)dxj1 ∧ . . . ∧ dxjn2(D.1.53)

Let the exterior multiplication ∧φbe with respect to the bilinear mapping:

φ : X1 ×X2 → X, (D.1.54)

where X1, X2 are the spaces of values of a, b respectively. Then a∧φb is given by:

a∧φb = φ

(α(x), β(x)

)dxi1 ∧ . . . ∧ dxin1

∧ dxj1 ∧ . . . ∧ dxjn2. (D.1.55)

In (D.1.55) the factor φ

(α(x), β(x)

)is known and the factor

dxi1 ∧ . . . ∧ dxin1∧ dxj1 ∧ . . . ∧ dxjn2

(D.1.56)

129


can be separated into two cases:

a) When dxi1 , . . . , dxin1, dxj1 , . . . , dxjn2

are not distinct.

Then Theorem D.1.3 gives that (D.1.55) is 0.

b) When dxi1 , . . . , dxin1, dxj1 , . . . , dxjn2

are distinct.

Now choose the permutation σ such that i1, . . . , in1, ji, . . . , jn2

are in strictlyincreasing order

k1, k2, . . . , kn1+n2 (D.1.57)

and thus

dxi1 ∧ . . .∧dxin1∧dxj1 ∧ . . .∧dxjn2

= sgn(σ)dxk1 ∧ . . .∧dxkn1+n2(D.1.58)

Finally what is left is to calculate the exterior dierential of a dierential formrepresented canonically. Once more consider a dierential form consisting ofonly one term which can be done without loss of generality.

Theorem D.1.6. [Car06] Given a dierential form:

µ = c(x)dxi1 ∧ . . . ∧ dxin , (D.1.59)

where c is a mapping U → X of class C1. Then:

dµ = dc ∧ dxi1 ∧ . . . ∧ dxin . (D.1.60)

Proof. If µ consists of dierential forms µ1, . . . , µn, where µi is an ni-form, suchthat

µ = µ1 ∧ . . . ∧ µm. (D.1.61)

Then utilizing Theorem D.1.2 repeatedly on (D.1.60) gives:

dµ = dµ1 ∧ µ2 ∧ . . . ∧ µm + (−1)n1µ1 ∧ dµ2 ∧ . . . ∧ µm+

+ . . .+ (−1)n1+...+nm−1µ1 ∧ . . . ∧ µm−1 ∧ dµm. (D.1.62)

Now applying (D.1.62) on (D.1.60) gives the following due to c(x)dxi1 = c(x)∧

130


dxi1 :

dµ = dc ∧ dxi1 ∧ . . . ∧ dxin + c d(dxi1) ∧ dxi2 ∧ . . . ∧ dxin+

+ . . .± c dxi1 ∧ . . . ∧ dxin−1 ∧ d(dxin). (D.1.63)

and using Theorem D.1.3 gives that:

d(dxi1) = 0, . . . , d(dxin) = 0 (D.1.64)

and thus:

dµ = dc ∧ dxi1 ∧ . . . ∧ dxin . (D.1.65)

131

E FUNCTIONS

E Functions

In this section some elementary notions about functions will be covered. Exam-ple of subject covered are domains, codomains of function, continuously dier-entiable functions and bijective functions.

Denition E.0.7 (Domain of function). Given a function f : X → Y . Thedomain of f is X (see Figure 15).

Figure 15: Domain and codomain of a function f

Example E.0.8 (Domain of function). Consider the function f : R→ R, givenby: f(x) = x2. The domain of f is R

Denition E.0.8 (Codomain of function). Given a function f : X → Y . Thecodomain of f is Y (see Figure 15).

Example E.0.9 (Codomain of function). Consider the function f : R → R,given by:

f(x) = x2. (E.0.66)

The codomain of f is R

Example E.0.10 (Codomain of function). Consider the function f : R→ R+,given by:

f(x) = x2. (E.0.67)

The codomain of f is R+

133

E FUNCTIONS

When dealing with expressions concerning partial derivatives, the expressionseasily gets long and complicated so before dening dierentiability, the conceptof multi-indices will be introduced to simplify the problem with partial deriva-tives.

Denition E.0.9 (Multi-index). An n-dimensional multi-index α is an n-tupleα = [α1, ..., αn] where α1, ..., αn ∈ N0.

The following operations for multi-indices are dened below as properties whichsatisfy for all α, β ∈ Nn0 :

Property 1 (Partial ordering). The partial ordering α ≤ β is equivalent to:

αi ≤ βi, (E.0.68)

for all i ∈ 1, ..., n.

Property 2 (Component-wise addition and subtraction). Addition and sub-traction of multi-indices are given by:

α± β = [α1 ± β1, ..., αn ± βn] (E.0.69)

and for the subtraction to be dened, α ≤ β needs to be satised.

Property 3 (Absolute value). The absolute value of α is given by:

|α| = α1 + ...+ αn. (E.0.70)

Property 4 (Factorial). The factorial of α is given by:

α! = α1! · ... · αn!. (E.0.71)

Property 5 (Power). The power α of x , xα is given by:

xα = xα11 · ... · xαnn . (E.0.72)

Example E.0.11 (Multi-index). Given the multi-indices

α = [1, 2, 3, 4, 0, 5] (E.0.73)

andβ = [0, 1, 3, 2, 0, 3]. (E.0.74)

134

E FUNCTIONS

Hereα+ β = [1, 3, 6, 6, 0, 8] (E.0.75)

and β ≤ α. Furthermore |α| = 15 and α! = 120

Denition E.0.10 (Continuously dierentiable). A function f : Rn → R is ktimes continuously dierentiable (class Ck) if all the partial derivatives:

Dαf =∂|α|f

∂xα11 · · · ∂x

αnn, |α| ≤ k (E.0.76)

exist and are continuous for all x = [x1, .., xn] ∈ Rn.

Remark E.0.6. A function which is innitely times continuously dierentiableis of class C∞ and is called smooth. Furthermore

C∞ ⊂ Ck+1 ⊂ Ck ⊂ C0. (E.0.77)

Remark E.0.7. A function f : Rm → Rn, m ≥ n is k-times continuously dier-entiable if every component of f = [f1, . . . , fn] is k-times continuously dieren-tiable.

Example E.0.12 (Continuously dierentiable). Given the multi-indexα = [1, 0, 0]. A partial derivative of f at x = [x1, x2, x3] is

Dαf =∂f

∂x1. (E.0.78)

Example E.0.13 (Continuously dierentiable). Given the multi-indexα = [1, 0, 1]. A partial derivative of f at x = [x1, x2, x3] is

Dαf =∂2f

∂x1∂x3. (E.0.79)

Example E.0.14 (Continuously dierentiable). The function f(x, y) = xy onR2 belongs to the classes

C0, C1, ..., C∞. (E.0.80)

Denition E.0.11 (Bijection). A function f : X → Y is a bijection if thefollowing is fullled for all x ∈ X and y ∈ Y .

a) Every element of X is mapped on exactlyone element of Y (Function with domain X)

135

E FUNCTIONS

b) Every element of Y was mapped on by at least one element of X (Surjection)

c) No element of Y was mapped on by more than one element of X (Injection)

Example E.0.15 (Bijection). The function f : R→ R, given by f(x) = x+ 1is a bijection. Condition a) is satised by choosing x ∈ R in the domain. Thiselement is mapped on the unique element x+1. To see that b) is satised, choosey ∈ R in the codomain. This element was mapped on by y − 1 and it is alsounique so c) is also fullled.

Example E.0.16 (Bijection). The function f : R → R, given by f(x) = x2

is not a bijection. Condition a) is satised by choosing x ∈ R in the domain.This element is mapped on the unique element x2. To see that b) is not satis-ed, choose for example −1 ∈ R in the codomain. This element was mapped onby any element in the domain so f is not surjective. Condition c) is also notfullled, for example choosing 1 ∈ R in the codomain. This element was mappedon by the two elements −1, 1 ∈ R in the domain so f is not injective.

Example E.0.17 (Bijection). The function f : X → Y where X = −1, 0, 1,Y = 0, 1, 2 and given by f(x) = x + 1 is a bijection. Condition a) is satis-ed by choosing x ∈ X in the domain. This element is mapped on the uniqueelement x + 1. To see that b) is satised, choose y ∈ Y in the codomain. Thiselement was mapped on by y − 1 and it is also unique so c) is also fullled.

Example E.0.18 (Bijection). The function f : X → Y where X = −1, 0, 1,Y = −1, 0, 1, 2 and given by f(x) = x + 1 is not a bijection. Condition a)is satised by choosing x ∈ X in the domain. This element is mapped on theunique element x + 1. To see that b) not is satised, choose −1 ∈ Y in thecodomain. This element was not mapped on by any element in the domain so fis not surjective. But c) is true, choose y ∈ Y \−1. This element was mappedon by the unique element y − 1 and the element −1 ∈ Y was not mapped on byany element so f is injective.

136

F EUCLIDEAN GEOMETRY

F Euclidean geometry

Denition F.0.12 (Euclidean structure). Given a vector space X. An Eu-clidean structure has the following forms for inner product, norm, angles andmetric for all vectors x, y ∈ X, where x = [x1, ..., xn] and y = [y1, ..., yn]:

a) 〈x, y〉 = x · y =n∑i=1

xiyi (Inner product)

b) ‖x‖ =√x · x (Norm)

c) The angle θ (0 ≤ θ ≤ π) is given by: (Angle)

θ = cos−1

(x · y‖x‖ · ‖y‖

)(F.0.81)

d) The metric:d(x, y) = ‖x− y‖ (Euclidean metric) (F.0.82)

Example F.0.19 (Euclidean structure). Given the vector space R. Then thefollowing is an Euclidean structure for all x, y ∈ R.

a) 〈x, y〉 = xy (Inner product)

b) ‖x‖ = |x| (Norm)

c) θ = cos−1

(xy|x|·|y|

)(Angle)

d) d(x, y) = |x− y| (Metric)

Denition F.0.13 (Euclidean space). An n-dimensional real coordinate spacetogether with an Euclidean structure is the n-dimensional Euclidean space whichwill be denoted En.

Example F.0.20 (Euclidean space). R together with the Euclidean structurein Example F.0.19 is the Euclidean space E.

137

G NOTATION

G Notation

The notation which will be used is presented below.

x, y, p, points in sets.

Ac, the complement of a set A.

n,m, dimensions in dierent spaces.

γ, a curve or path.

l(γ), the length of the curve γ.

l(γ(a), γ(b)), the length of the curve segment from γ(a) to γ(b).

Ω, a domain.

[a, b], notation for vectors.[a bc d

], notation for matrices.

AT , the transpose to a matrix A.

R, the real numbers.

R+, the non-negative real numbers.

R−, the non-positive real numbers.

C, the complex numbers.

Z, the integers.

Q, the rational numbers.

N, the positive integers not including 0.

N0, the positive integers including 0.

Re(z), is the real part x of z = x+ iy.

z, the conjugate of z.

f, g, functions and maps.

f g, is the composition f of g.

d, a metric or distance.

dI , an intrinsic metric.

139

G NOTATION

Bd(x, r), open ball with center at x, radius r and metric d.

Bd(x, r), closed ball with center at x, radius r and metric d.

B+(x, r), a forward metric ball centered at x with radius r.

X,Y , will usually be used to denote topological, metric, Hausdor, vector , in-ner product spaces or other spaces.

Xk, a sequence X0, X1, X2, . . ..

limn→∞

Xk, the limit of the sequence Xk.

σ, a topology.

U, V,W , neighborhoods of points.

(G, •) or G, a group G with operation •.

(F,+, •) or F , a eld F with operations +, •.

∼, an equivalence relation.

[a]X , the equivalence class of a in X.

L, a linear map.

E, a basis.

X∗, the dual space of X.

〈x, y〉, an inner product.

‖·‖, a norm.

α, β, multi-indices.

Ck, the class of k-times continuously dierentiable functions.

Ed, an n-dimensional Euclidean space.

I, an index set.

I, an interval.

I[a,b], the closed interval between a and b.

I(a,b)], the open interval between a and b.

Dfp, the dierential map of f at p.

dfp, the dierential of f at p.

140

G NOTATION

p, a point in a manifold.

M,N , for manifolds and submanifolds.

D2, the 2-dimensional unit disc.

Sn, the n-dimensional sphere.

Ckp (M), the germs of functions of the class Ck on the set M at p.

f ≡ c, the function f : X → Y is for every x ∈ X, f(x) = c where c ∈ R.

TpM , a tangent space at p for a manifold M .

TM , the tangent bundle for a manifold M .

B, a bilinear form.

f |X = g|X , the functions satises f(x) = g(x) for every x ∈ X.

g, a Riemannian metric.

[X,Y ], the Lie bracket of X,Y .

[X,Y ]p, the Lie bracket of X,Y at a point p.

Dxf , the directional derivative of f at the direction x.

F , a Finsler structure.

SM , the sphere bundle of M .

A, the Cartan tensor in Finsler geometry.

N ij , the non-linear connection.

`, a distinguished section.

ω, a Hilbert form.

Λ, a linear connection.

An(X,Y ), the space of n-linear continuous mapping between Xn → Y .

Ln(X,Y ), the space of n-linear mappings between Xn → Y .

Ω(k)n (U, Y ), the space of n-forms dened in U with values at Y and of class Ck.

da, the exterior dierentiation of a dierential form a.

141

REFERENCES

H Bibliography

References

[AR10] H. Anton and C. Rorres. Elementary Linear Algebra. John Wiley &Sons Canada, Limited, 2010.

[BBI01] Dmitri Burago, Yuri Burago, and Sergei Ivanov. A course in metricgeometry, volume 33 of Graduate Studies in Mathematics. AmericanMathematical Society, Providence, RI, 2001.

[BCS00] D. Bao, S.-S. Chern, and Z. Shen. An introduction to Riemann-Finslergeometry, volume 200 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2000.

[BJ82] Theodor Bröcker and Klaus Jänich. Introduction to dierential topol-ogy. Cambridge University Press, Cambridge, 1982. Translated fromthe German by C. B. Thomas and M. J. Thomas.

[Car99] Élie Cartan. Sur certaines expressions diérentielles et le problème dePfa. Ann. Sci. École Norm. Sup. (3), 16:239332, 1899.

[Car06] Henri Cartan. Dierential forms. Dover publications,Inc., Mineola,New York, 2006.

[Fra67] John B. Fraleigh. A rst course in abstract algebra. Addison-WesleyPublishing Co., Reading, Mass.-London-Don Mills, Ont., 1967.

[GHL04] Sylvestre Gallot, Dominique Hulin, and Jacques Lafontaine. Rieman-nian geometry. Universitext. Springer-Verlag, Berlin, third edition,2004.

[Gil91] George T. Gilbert. Positive denite matrices and Sylvester's criterion.Amer. Math. Monthly, 98(1):4446, 1991.

[HJ85] Roger A. Horn and Charles R. Johnson. Matrix analysis. CambridgeUniversity Press, Cambridge, 1985.

[Jos11] Jürgen Jost. Riemannian geometry and geometric analysis. Universi-text. Springer, Heidelberg, sixth edition, 2011.

[Kaw56] Akitsugu Kawaguchi. On the theory of non-linear connections. II.Theory of Minkowski spaces and of non-linear connections in a Finslerspace. Tensor (N.S.), 6:165199, 1956.

[Kik62] Shigetaka Kikuchi. Theory of Minkowski space and non-linear con-nections in a Finsler space. Tensor (N.S.), 12:4760, 1962.

[KN63] Shoshichi Kobayashi and Katsumi Nomizu. Foundations of dierentialgeometry. Vol I. Interscience Publishers, a division of John Wiley &Sons, New York-Lond on, 1963.

[Küh02] Wolfgang Kühnel. Dierential geometry, volume 16 of Student Math-ematical Library. American Mathematical Society, Providence, RI,2002. Curvessurfacesmanifolds, Translated from the 1999 Ger-man original by Bruce Hunt.

143

REFERENCES REFERENCES

[Mat86] Makoto Matsumoto. Foundations of Finsler geometry and specialFinsler spaces. Kaiseisha Press, Shigaken, 1986.

[Mor05] Frank Morgan. Real analysis and applications. American Mathemat-ical Society, Providence, RI, 2005. Including Fourier series and thecalculus of variations.

[MVJ10] F.P. Miller, A.F. Vandome, and M.B. John. Lipschitz Continuity.VDM Verlag Dr. Mueller e.K., 2010.

[NSS11] R.K. Nagle, E.B. Sa, and A.D Snider. Fundamentals of DierentialEquations and Boundary Value Problems. Pearson Education, 2011.

[Rud76] Walter Rudin. Principles of mathematical analysis. McGraw-Hill BookCo., New York, third edition, 1976. International Series in Pure andApplied Mathematics.

144

Index

Abelian group, 105Absolute homogeneous, 66Abstract manifold, 32Abstract submanifold, 32Accumulation point, 75, 81Alternating multilinear form, 117Arzela-Ascoli theorem, 12Atlas, 28

Ball, 67, 73Base, 92Basis, 111, 125Bijection, 27, 90, 131Bilinear form, 43, 119Binary operator, 103, 110Binary relation, 103Bolzano-Weierstrass theorem, 89Bounded, 73Bounded sequence, 12Boundedly compact, 15, 56

Canonical projection map, 61Cantor´s diagonalization argument, 90Cartan tensor, 61Cartesian product, 35, 103Cauchy sequence, 13, 74Cauchy-Schwarz inequality, 114Chart, 32Chern connection, 64Christoel symbols, 52, 61Clopen set, 82Closed, 77Closed set, 15, 82Codomain of function, 129Compact, 14, 67, 84Complete, 19, 56, 75Connected, 5, 54, 69, 79Continuous, 27, 76, 80Continuous map, 2Continuously dierentiable, 25, 131Convex, 5, 101Cotangent space, 62Countable, 13, 90Covariant derivatives, 63Covector, 59Curve, 2, 99

Dense, 13, 91Derivation, 39Determinant, 59, 107, 118

Dieomorphism, 28, 67Dierential form, 121Dierential map, 25Dilation, 76Discrete metric, 71, 85Discrete topology, 27, 78, 92, 99, 100Distinguished section, 62Distributive, 106Domain, 8, 100Domain of function, 129Dual basis, 44, 112Dual space, 44, 112

Einstein summation convention, 59Equivalence class, 32, 34, 104Equivalence relation, 34, 104Equivalent atlases, 32Equivalent paths, 3Euclidean inner product, 133Euclidean metric, 133Euclidean norm, 133Euclidean space, 5, 133Euclidean structure, 133Exterior dierentiation, 123Exterior product, 120, 122

Field, 106, 110Finsler manifold, 57Finsler structure, 58First variation of arc length, 66Forward bounded, 69Forward boundedly compact, 69Forward Cauchy sequence, 69Forward complete, 69Forward metric ball, 68Fundamental tensor, 61

Geodesic, 19, 53, 66Germs, 36Group, 105

Hausdor space, 16, 28, 100Heine-Borel theorem, 15, 87Hermitian matrix, 59, 109Hilbert form, 62Homeomorphism, 27Hopf-Rinow-Cohn-Vossen theorem, 20,

56, 69Horizontal subspace, 63

145

INDEX INDEX

Inner product, 43, 113Inner product space, 114Interior point, 84Intrinsic metric, 4, 54, 66

Jacobi identity, 48

Koszul formula, 50

Lebesgue number, 98Lebesgue´s Lemma, 98Length of curve, 54, 65, 99Length space, 5, 54, 66Lie bracket, 47Limit, 75, 82Lindelöf compactness, 95Linear connection, 64Linear map, 39, 111Linearly dependent, 118Linearly independent, 111Lipschitz constant, 76Lipschitz continuous, 10, 75, 99Locally compact, 15, 56Locally Minkowskian manifold, 58Lorentzian metric, 45

Maximal atlas, 31Metric, 55, 66, 71Metric on Finsler manifold, 66Metric on Riemannian manifold, 54Metric space, 14, 71, 72Minimal geodesic, 20Minkowski norm, 58Multi-index, 130

n-linear continuous alternating mappings,119

Natural curve, 6, 53Natural projection, 57Neighborhood, 15, 26, 81Net, 16Non-expanding, 76Non-linear connection, 62Norm, 58, 116

Open set, 73, 78

Parametrization, 3Path, 2Path-connected, 79Permutation, 107, 119Piecewise smooth variation, 65Pointwise convergence, 7Positive-denite, 45, 58, 108

Precompact, 22, 68Pulled-back bundle, 61

Real coordinate space, 71Rectiable, 99Rectiable curve, 10Riemannian connection, 49Riemannian Finsler manifold, 58Riemannian manifold, 46, 58Riemannian metric, 45, 58

Sasaki metric, 63Semi-metric, 67Semi-Riemannian metric, 45Separable, 92Sequence, 74Sequentially compact, 17, 89, 96Sesquilinear, 44Shortest path, 4, 56, 70Sphere bundle, 60Standard topology, 15, 29, 79Subeld, 106Submanifolds of Euclidean spaces, 26Submersion, 25Surjection, 132Sylvester´s criterion, 59, 109

t-curve, 65Tangent bundle, 35, 57Tangent in abstract manifold, 34, 40Tangent in submanifold, 33Tangent space, 58Tangent space in abstract manifold, 34Tangent space in submanifold, 33Tensor, 61, 112Topological space, 2, 36, 77Topology, 55, 68, 78Triangle inequality, 71, 116Trivial topology, 80, 100

u-curve, 65Uniform convergence, 7Uniform convergence of curves, 8Uniformly continuous, 76

Vector, 59, 110Vector eld, 36Vector space, 41, 43, 100, 110Vertical subspace, 63

146

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