Diff Geom Notes.pdf

8/14/2019 Diff Geom Notes.pdf

1/219

Lecture NotesIntroduction to Differential Geometry

MATH 442

Instructor: Ivan Avramidi

New Mexico Institute of Mining and Technology

Socorro, NM 87801

August 25, 2005

Author: Ivan Avramidi;File: diffgeom.tex;Date: April 12, 2006;Time: 17:59


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Contents

1 Manifolds 1

1.1 Submanifolds of Euclidean Space . . . . . . . . . . . . . . . . . 11.1.1 Submanifolds ofRn . . . . . . . . . . . . . . . . . . . . . 2

1.1.2 Differential of a Map . . . . . . . . . . . . . . . . . . . . 4

1.1.3 Main Theorem on Submanifolds ofRm . . . . . . . . . . 5

1.2 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.1 Basic Notions of Topology . . . . . . . . . . . . . . . . . 6

1.2.2 Idea of a Manifold . . . . . . . . . . . . . . . . . . . . . 9

1.2.3 Rigorous Definition of a Manifold . . . . . . . . . . . . . 11

1.2.4 Complex Manifolds . . . . . . . . . . . . . . . . . . . . 12

1.3 Tangent Vectors and Mappings . . . . . . . . . . . . . . . . . . . 14

1.3.1 Tangent Vectors . . . . . . . . . . . . . . . . . . . . . . . 14

1.3.2 Vectors as Differential Operators . . . . . . . . . . . . . . 15

1.3.3 Tangent Space . . . . . . . . . . . . . . . . . . . . . . . 16

1.3.4 Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.3.5 Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . 18

1.3.6 Change of Coordinates . . . . . . . . . . . . . . . . . . . 20

1.4 Vector Fields and Flows . . . . . . . . . . . . . . . . . . . . . . . 21

1.4.1 Vector Fields in Rn . . . . . . . . . . . . . . . . . . . . . 21

1.4.2 Vector Fields on Manifolds . . . . . . . . . . . . . . . . . 23

1.4.3 Straightening Flows . . . . . . . . . . . . . . . . . . . . 23

2 Tensors 252.1 Covectors and Riemannian Metric . . . . . . . . . . . . . . . . . 25

2.1.1 Linear Functionals and Dual Space . . . . . . . . . . . . 25

2.1.2 Differential of a Function . . . . . . . . . . . . . . . . . . 27

2.1.3 Inner Product . . . . . . . . . . . . . . . . . . . . . . . . 28

2.1.4 Riemannian Manifolds . . . . . . . . . . . . . . . . . . . 31

I


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II CONTENTS

2.1.5 Curves of Steepest Ascent . . . . . . . . . . . . . . . . . 32

2.2 Tangent Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.2.1 Fiber Bundles . . . . . . . . . . . . . . . . . . . . . . . . 33

2.2.2 Tangent Bundle . . . . . . . . . . . . . . . . . . . . . . . 35

2.3 The Cotangent Bundle . . . . . . . . . . . . . . . . . . . . . . . 38

2.3.1 Pull-Back of a Covector . . . . . . . . . . . . . . . . . . 39

2.3.2 Phase Space . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.3.3 The Poincare 1-Form . . . . . . . . . . . . . . . . . . . . 42

2.4 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.4.1 Covariant Tensors . . . . . . . . . . . . . . . . . . . . . 44

2.4.2 Contravariant Tensors . . . . . . . . . . . . . . . . . . . 46

2.4.3 General Tensors of Type (p, q) . . . . . . . . . . . . . . . 482.4.4 Linear Transformations and Tensors . . . . . . . . . . . . 50

2.4.5 Tensor Fields . . . . . . . . . . . . . . . . . . . . . . . . 51

2.4.6 Tensor Bundles . . . . . . . . . . . . . . . . . . . . . . . 51

2.4.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.4.8 Einstein Summation Convention . . . . . . . . . . . . . . 53

3 Differential Forms 55

3.1 Exterior Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.1.1 Permutation Group . . . . . . . . . . . . . . . . . . . . . 55

3.1.2 Permutations of Tensors . . . . . . . . . . . . . . . . . . 57

3.1.3 Alternating Tensors . . . . . . . . . . . . . . . . . . . . . 583.1.4 Exteriorp-forms . . . . . . . . . . . . . . . . . . . . . . 62

3.1.5 Exterior Product . . . . . . . . . . . . . . . . . . . . . . 64

3.1.6 Interior Product . . . . . . . . . . . . . . . . . . . . . . . 66

3.2 Orientation and the Volume Form . . . . . . . . . . . . . . . . . 68

3.2.1 Orientation of a Vector Space . . . . . . . . . . . . . . . 68

3.2.2 Orientation of a Manifold . . . . . . . . . . . . . . . . . 69

3.2.3 Hypersurfaces in Orientable Manifolds . . . . . . . . . . 70

3.2.4 Projective Spaces . . . . . . . . . . . . . . . . . . . . . . 71

3.2.5 Pseudotensors and Tensor Densities . . . . . . . . . . . . 73

3.2.6 Volume Form . . . . . . . . . . . . . . . . . . . . . . . . 753.2.7 Star Operator and Duality . . . . . . . . . . . . . . . . . 77

3.3 Exterior Derivative . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.3.1 Coderivative . . . . . . . . . . . . . . . . . . . . . . . . 82

3.4 Pullback of Forms . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.5 Vector Analysis in R3 . . . . . . . . . . . . . . . . . . . . . . . . 85

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CONTENTS III

3.5.1 Vector Algebra in R3 . . . . . . . . . . . . . . . . . . . . 85

3.5.2 Vector Analysis in R3

. . . . . . . . . . . . . . . . . . . . 86

4 Integration of Differential Forms 89

4.1 Integration over a Parametrized Subset . . . . . . . . . . . . . . . 89

4.1.1 Integration ofn-Forms in Rn . . . . . . . . . . . . . . . . 89

4.1.2 Integration over Parametrized Subsets . . . . . . . . . . . 90

4.1.3 Line Integrals . . . . . . . . . . . . . . . . . . . . . . . . 91

4.1.4 Surface Integrals . . . . . . . . . . . . . . . . . . . . . . 91

4.1.5 Independence of Parametrization . . . . . . . . . . . . . . 92

4.1.6 Integrals and Pullbacks . . . . . . . . . . . . . . . . . . . 93

4.2 Integration over Manifolds . . . . . . . . . . . . . . . . . . . . . 954.2.1 Partition of Unity . . . . . . . . . . . . . . . . . . . . . . 95

4.2.2 Integration over Submanifolds . . . . . . . . . . . . . . . 97

4.2.3 Manifolds with boundary . . . . . . . . . . . . . . . . . . 98

4.3 Stokess Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.3.1 Orientation of the Boundary . . . . . . . . . . . . . . . . 99

4.3.2 Stokes Theorem . . . . . . . . . . . . . . . . . . . . . . 99

4.4 Poincare Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5 Lie Derivative 107

5.1 Lie Derivative of a Vector Field . . . . . . . . . . . . . . . . . . . 107

5.1.1 Lie Bracket . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.1.2 Flow generated by the Lie Bracket . . . . . . . . . . . . . 110

5.2 Lie Derivative of Forms and Tensors . . . . . . . . . . . . . . . . 112

5.2.1 Properties of Lie Derivative . . . . . . . . . . . . . . . . 114

5.3 Frobenius Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 1 19

5.3.1 Distributions . . . . . . . . . . . . . . . . . . . . . . . . 119

5.3.2 Frobenius Theorem . . . . . . . . . . . . . . . . . . . . . 123

5.3.3 Foliations . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5.4 Degree of a Map . . . . . . . . . . . . . . . . . . . . . . . . . . 1285.4.1 Gauss-Bonnet Theorem . . . . . . . . . . . . . . . . . . 128

5.4.2 Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . 130

5.4.3 Brouwer Degree . . . . . . . . . . . . . . . . . . . . . . 133

5.4.4 Index of a Vector Field . . . . . . . . . . . . . . . . . . . 137

5.4.5 Linking Number . . . . . . . . . . . . . . . . . . . . . . 140



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IV CONTENTS

6 Connection and Curvature 143

6.1 Affine Connection . . . . . . . . . . . . . . . . . . . . . . . . . . 1436.1.1 Covariant Derivative . . . . . . . . . . . . . . . . . . . . 143

6.1.2 Curvature, Torsion and Levi-Civita Connection . . . . . . 146

6.1.3 Parallel Transport . . . . . . . . . . . . . . . . . . . . . . 149

6.2 Tensor Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

6.2.1 Covariant Derivative of Tensors . . . . . . . . . . . . . . 150

6.2.2 Ricci Identities . . . . . . . . . . . . . . . . . . . . . . . 1 51

6.2.3 Normal Coordinates . . . . . . . . . . . . . . . . . . . . 152

6.2.4 Properties of the Curvature Tensor . . . . . . . . . . . . . 153

6.2.5 Bianci Identities . . . . . . . . . . . . . . . . . . . . . . 155

6.3 Cartans Structural Equations . . . . . . . . . . . . . . . . . . . . 157

7 Homology Theory 163

7.1 Algebraic Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 163

7.1.1 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

7.1.2 Finitely Generated and Free Abelian Groups . . . . . . . 167

7.1.3 Cyclic Groups . . . . . . . . . . . . . . . . . . . . . . . 167

7.2 Singular Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 69

7.2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 175

7.3 Singular Homology Groups . . . . . . . . . . . . . . . . . . . . . 176

7.3.1 Cycles, Boundaries and Homology Groups . . . . . . . . 1767.3.2 Simplicial Homology . . . . . . . . . . . . . . . . . . . . 177

7.3.3 Betti Numbers and Topological Invariants . . . . . . . . . 178

7.3.4 Some Theorems from Algebraic Topology . . . . . . . . . 180

7.3.5 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . 184

7.4 de Rham Cohomology Groups . . . . . . . . . . . . . . . . . . . 186

7.4.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 189

7.5 Harmonic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . 190

7.6 Relative Homology and Morse Theory . . . . . . . . . . . . . . . 197

7.6.1 Relative Homology . . . . . . . . . . . . . . . . . . . . . 197

7.6.2 Morse Theory . . . . . . . . . . . . . . . . . . . . . . . . 201

8 207

8.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

Bibliography 209



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CONTENTS V

Notation 211



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VI CONTENTS



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Chapter 1

Manifolds

1.1 Submanifolds of Euclidean Space

Idea: Manifold is a general space that looks locally like a Euclidean spaceof the same dimension. This allows to develop the differential and integral

calculus.

LetnN be a positive integer. TheEuclidean space Rn is a set of pointsxdescribed by orderedn-tuples (x1, . . . ,xn) or real numbers.

The numbers xi R, i = 1, . . . , n, are called the Cartesian coordinatesofthe point x.

The integern is thedimensionof the Euclidean space.

Thedistancebetween two points of the Euclidean space is defined by

d(x,y) =

nk=1

(xk yk)2 .

Theopen ballof radius centered at x0is the set of points defined byB(x0) ={xRn |d(x,x0)< } .

Aneighborhoodof a point x0 is the set of points that contain an open ballaround it.

1


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2 CHAPTER 1. MANIFOLDS

Let x0 Rn be a fixed point with Cartesian coordinates xi, i = 1, . . . , n,in the Euclidean space and S R

n

be a neighborhood ofx0. An injective(one-to-one) map

f :S Rn

defined by

yi = fi(x1, . . . ,xn) , i =1, . . . , n,

where fi(x) are smooth functions, is called a coordinate systemin S.

The map fis injective if for any point xin S

det fixj 0 .1.1.1 Submanifolds ofRn

Letn, r N be positive integers andm = n + r. Let M Rm be a subset ofthe Euclidean space Rm.

ThenMis asubmanifoldofRm of dimensionnif for any pointx0 Min Mthere exists a neighborhood with a coordinate system such that every point

in this neighborhood has coordinates (x1, . . . ,xn,xn+1, . . . ,xn+r), where the

last rcoordinates (xn+

1, . . . ,xn+

r) are given by smooth functions of the firstncoordinates (x1, . . . ,xn):

x = f(x1, . . . ,xn) , =n + 1, . . . , n + r.

The coordinates (x1, . . . ,xn) are called thelocal coordinatesfor Mnearx0.

More generally, letF :Rm Rr

be a smooth map described byrequations

y = F(x1, . . . ,xm) , =1, . . . , r.

Lety0Rr and

M = F1(y0) ={xRm | F(x) =y0}



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1.1. SUBMANIFOLDS OF EUCLIDEAN SPACE 3

be a subset of the Euclidean space Rm described by the locus ofrequations

F(x1, . . . ,xm) = y0, =1, . . . , r.

Suppose that Mis non-empty and let x0 M, that isF(x0) =y0.

Then theImplicit Function Theoremsays that if

det

F

x(x0)

0 ,

where =1, . . . , rand =n + 1, . . . , n + r, then there is a neighborhood of

x0 such that the last rcoordinates can be expressed as smooth functions ofthe firstn coordinates:

x = f(x1, . . . ,xn) , =n + 1, . . . , n + r.

If this is true for any point ofM, then Mis an-dimensional submanifold ofRm.

The matrix F

xj

,

where =

1, . . . , rand j =

1, . . . , mis called theJacobian matrix.

TheGeneral Implicit Function Theorem says that if at a point x0 the Ja-cobian matrix has the maximal rank equal to r,

rank

F

xj(x0)

=r,

then there exists a coordinate system in a neighborhood ofx0 such that the

lastrcoordinates can be expressed as smooth functions of the first n coor-

dinates.

If this is true for every point ofM, then Mis an-dimensional submanifoldofRm. The numberris called the codimensionofM. If the codimension r is equal to 1, that is n = m1, then M is called a

hypersurface.



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1.1.2 Differential of a Map

Let Rn be a Euclidean space and x0 Rm. Then thetangent spaceto Rm atx0 is a vector space R

mx of all vectors in R

m based atx.

LetF :Rm Rr

be a smooth map described byrsmooth functions

y = F(x1, . . . ,xn) , =1, . . . , r.

Letx0Rm andx = x(t),t(, ), be a curve in Rm such that

x(0) = x0anddx

dt(0) =v ,

wherevRmx0 is the tangent vector to the curve at x0.

Lety0 = F(x0)Rr andy(t) = F(x(t)). Then

y(0) =y0anddy

dt

(0) =w ,

wherewRrx0 is the tangent vector to the image of the curve at y0.

We compute

w =

mi=1

F

xi

(x0)v

i .

Thus there is a linear transformation

F :Rmx0 R

ry0 ,

so that

Fv =w .

Fis called thedifferentialof the map Fat x0.



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1.1. SUBMANIFOLDS OF EUCLIDEAN SPACE 5

1.1.3 Main Theorem on Submanifolds ofRm

The matrix of the linear transformationFis exactly the Jacobian matrix. Therefore, the differentialF at a point x0is asurjective (onto)map if and

only ifmrand the Jacobian atx0has the maximal rank equal to r. Recall that F : Rmx0 Rry0 is surjective if for any w Rry0 there isv Rmx0

such thatFv =w.

Thus, we have the following theorem.

Theorem 1.1.1 Let F :Rm Rr with m> r, y0Rr and

M = F1(y0) ={xRm | F(x) =y0} .

If M is non-empty and for any x0 M the differential F :Rmx0 Rry0 issurjective, then M is a n =(m r)-dimensional submanifold ofRm.



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1.2 Manifolds

1.2.1 Basic Notions of Topology

First we define the basic topological notions in the Euclidean space Rn.

Letx0Rn be a point in Rn and >0 be a positive real number.

Theopen ballin Rn of radius with the center at x0is the set

B(x0) ={xRn |d(x,x0)< } .

Theclosed ballin Rn of radius with the center at x0is the set

B(x0) ={xRn |d(x,x0)} .

LetU Rn be a subset ofRn. A point x U is an interior point ofUif there is an open ball B(x) of some radius > 0 centered at xsuch that

B(x)U.

A point x Rn is a boundary point ofUif every open ball B(x) of anyradius > 0 centered at xcontains at least one point from Uand one pointfrom its complement (that is not fromU).

The setUo of all interior points ofUis called theinteriorofU.

The setUof all boundary points ofUis called theboundaryofU.

A setU Rn isopenif every point ofUis an interior point ofU, that is,U =Uo.

A setF

Rn isclosedif its complementRn

\Fis open, that is,F = Fo

F.

The sets and Rn are both open and closed.

The union ofanycollection of open sets is open.

The intersection of anyfinitenumber of open sets is open.



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1.2. MANIFOLDS 7

Definition 1.2.1 A generaltopological spaceis a set M together with

a collection of subsets of M, calledopen sets, that satisfy the followingproperties

1. M and are open,2. the intersection of any finite number of open sets is open,

3. the union of any collection of open sets is open.

Such a collection of open sets is called a topologyof M.

A subset ofMis closedif its complementM

\Fis open.

Theclosure Sof a subsetS Mof a topological spaceMis the intersec-tion of all closed sets that contain S; it is equal to S = S S.

A subset S Mof a topological space is dense in M ifS = M, that is,every non-empty subset ofMcontains an element ofS.

A topological space is called separable if it contains a countable densesubset.

A topology on Mnaturally induces a topology on any subset of M. Let

A Mbe a subset of M. Then theinduced topology on A is defined asfollows. A subsetV Ais defined to be open subset ofA if there is an opensubsetU MofMsuch thatV =UA.

LetxMbe a point in a topological space M. An open set inMcontainingthe point xis called aneighborhoodofx.

Definition 1.2.2 Let M and N be two topological spaces and F : MN be a map from M into N. The map F is said to becontinuousif the

inverse image of any open set in N is an open set in M.

That is, if for any open set V N the setF1(V) ={xM|F(x)V}

is open in M.

The direct images of open sets do not have to be open!



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Let the map F be bijective, that is, injective (one-to-one) and surjective(onto). Then there exists the inverse map F

1

:N M.

Definition 1.2.3 A map F : MN is called ahomeomorphismif itis bijective and both F and F1 are continuous.

Homeomorphisms preserve topology, that is, they take open sets to opensets and closed sets to closed sets.

A topological space M is called Hausdorff if any two points of M havedisjoint neighborhoods.

A collection of subsets of a topological space Mis called anopen coverof

Mif the union of all subsets in the collection coincides with M.

A subcollection of subsets that is itself a cover is called asubcover.

Definition 1.2.4 A topological space M is called compact if everyopen cover of M has a finite subcover.

A subset MRn of a Euclidean space if called boundedif there is an openball BC(0) of some radiusCcentered at the origin such that M BC(0).

Theorem 1.2.1 Bolzano-Weierstrass Theorem. Let M Rn be asubset ofR

n

with the induced topology. Then M is compact if and onlyif M is closed and bounded in Rn.

Properties of Continuous Maps.

Theorem 1.2.2 Let M and N be two topological spaces and M be

compact. Let F : M N be a continuous map from M into N . Thenthe image F(M)of M is compact in N.

That is, a continuous image of a compact topological space is compact.

Proof:

1. Let {U}Abe an open cover ofF(M) inN.2. Then {F1(U)}Ais an open cover ofM.3. SinceMis compact it has a finite subcover {F1(Ui)}ni=1.4. Then {Ui}ni=1 is a finite subcover ofF(M).



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1.2. MANIFOLDS 9

5. ThusF(M) is compact.

Theorem 1.2.3 A continuous real-valued function f : M R on acompact topological space M is bounded.

Proof:

1. f(M) is compact in Rn.

2. Thus f(M) is closed and bounded.

1.2.2 Idea of a Manifold

A manifold Mof dimensionn is a topological space that is locally homeo-morphic to Rn.

A manifoldMis covered by a family of local coordinate systems {U;x1, . . . ,xn}A,called anatlas, consisting of open sets, calledpatches(orcharts),U, and

coordinatesx.

A point p

U

U that lies in two coordinate patches has two sets of

coordinatesxand x related by smooth functions

xi = fi

(x1

, . . . ,xn

) , i =1, . . . , n .

The coordinates xand x are said to becompatible.

If all the functions fare smooth, then the manifoldMis said to be smooth.If these functions are analytic, then the manifold is said to be real analytic.

Each patch is homeomorphic to some open subset in Rn.

Thus, a manifold is locally Euclidean.

The collection of all patches (charts) is called an atlas.

The collection of all coordinate systems that are compatible with those usedto define a manifold is called the maximal atlas.



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LetMbe an n-dimensional manifold with local coordinate systems {U;x1, . . . ,xn}AandNbe an m-dimensional manifold with local coordinate systems {V;y

1, . . . ,y

m }B.

Theproduct manifoldL = M Nis a manifold

L = M N ={(p, q)|pM, qN}

with local coordinate systems {W;z1, . . . ,zn+m }A,Bwhere

W =U Vand

(z1, . . . ,zn) =(x

1, . . . ,x

n)

and(zn+1 , . . . ,z

n+m ) =(y

1, . . . ,y

m ) .

Examples

Unit SphereSn.Sn ={xRn+1 |d(x, 0) =1}

Stereographic projection:

: Rn+1 Rn

Let

R =

1

nj=1

(xj)2

Then

xn+1 =R , 1,2(x) =

x1

1 R , , xn

1 R

.

TorusTn =S1 S1.

Tn

has local coordinates (1

, , n

) (angles). Topologically it is the cube [0, 1]n with the antipodal points identified.

Real Projective Space RPn.

RPn is the space of unoriented lines through the origin ofRn+1.



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1.2. MANIFOLDS 11

The set of oriented lines through the origin ofRn+1 isSn.

Topologically RPn is the sphere Sn in Rn+1 with the antipodal pointsidentified, which is the unit ball in Rn with the antipodal points on the

boundary (which is a unit sphereSn1) identified.

RPn is covered by (n + 1) sets

Uj ={LRPn | Lwithxj 0} , j =1, . . . , n + 1 .

The local coordinates inUjare

v1 = x1

xj,

, vn = xn

xj .

The (n + 1)-tuple (x1, . . . ,xn+1) identified with (x1, . . . , xn+1), 0,

are calledhomogeneous coordinatesof a point in RPn.

1.2.3 Rigorous Definition of a Manifold

Let Mbe a set (without topology) and{U}A be a collection of subsetsthat is a cover ofM, that is,

A U = M. Let

:URn, A ,be injective maps such that (U) are open subsets in R

n.

The set(U U) is an open subset in Rn. The maps

f = 1 : (U U)R

are called thetransition functions(or theoverlap functions). We assumethat the transition functions are smooth.

The pair (U, ) is called acoordinate patch(orchart). A point pUis assigned coordinates of the point (p) in Rn. Thus,is

called acoordinate map.



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Now, we take the maximal atlas of such coordinate patches. Thetopologyin Mis defined as follows. LetW Mbe a subset ofM. A point pW inWis said to be aninterior

point if there is a coordinate chart (U, ) including psuch thatUW. A subsetWofMis declared to beopenif all of its points are interior points. If the resulting topological space is Hausdorff and separable, then Mis said

to be ann-dimensionalsmooth manifold.

The regularity of the transition functions determines the regularity of the

manifold. If the transition functions are only differentiable once, then themanifold is calleddifferentiable. If they are of classCk, then the manifold

is called a manifoldof classCk. If they are of classC, then the manifoldis calledsmooth. If the transition functions are analytic, then the manifold

is calledanalytic.

Let F : M R be a real-valued function on M. Let (U,x) be a localcoordinate system. Then the function

F = F 1 : (U)R

is a function ofn real variables F(x1, . . . ,xn).

The function F is said to be smoothif the function F is smooth in termsof local coordinates x.

The process of replacing the map Fby the functionF = F 1 is usuallyomitted, and the functions FandF are identified, so that we think of the

functionFdirectly in terms of local coordinates.

1.2.4 Complex Manifolds

LetMbe a set and {U}Abe its cover. Let

:UCn

be injective maps from U to the complex n-space Cn so that (U) are

open subsets ofCn.



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1.2. MANIFOLDS 13

Let the transition functions

f = 1 : (U U)Cn ,

defined by

zk = fk

(z1

, . . . ,zn

) ,

wherezk = xk +iy

k andz

k

= xk +iyk

, k = 1, . . . , n, be complex analytic.

That is, they satisfy Cauchy-Riemann conditions

xk

xj

=yk

yj

,xk

yj

=yk

xj

,

orzk

zj

=0 ,

with j, k =1, . . . , n.

Then Mis called an-dimensionalcomplex manifold. The topological dimension of a n-dimensional complex manifold is 2n.



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1.3. TANGENT VECTORS AND MAPPINGS 15

Definition 1.3.1 A tangent vector at a point p0 M of a manifoldM is a map that assigns to each coordinate chart(U,x)about p0 anordered n-tuple(X1, . . . ,X

n)such that

Xi =

nj=1

xix

j

(p0)Xj.1.3.2 Vectors as Differential Operators

Let f : MR be a real-valued function on M.

LetpMbe a point on Mand Xbe a tangent vector at p. Let (U,x) be a coordinate chart about p. The (directional)derivative of fwith respect to Xat p(or along X, or in

the direction ofX) is defined by

Xp(f) = DX(f) =

nj=1

f

xj

(p)Xj.

Theorem 1.3.1 DX(f) does not depend on the local coordinate system.

Proof:

1. Chain rule.

Theintrinsicproperties areinvariantunder a change of coordinate system.They should not depend on the choice of the local chart.

There is a one-to-one correspondence between tangent vectors toMat p andfirst-order differential operators acting on real-valued functions in a local

coordinate chart (U,x) by

Xp =

nj=1

Xj

xj

p

.

Therefore, we can identify tangent vectors and the differential operators.



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Let us fix an index j =1, . . . , n. The curve

xi(t) = xi0, i j, xj(t) =t,

is called the j-thcoordinate curve.

The velocity vector to this curve is given by

xi(0) =ij,

whereijis theKronecker symboldefined by

i

j = 1, ifi = j ,0, ifi j .

The differential operator corresponding to this velocity vector is

xj

.

1.3.3 Tangent Space

LetMbe a manifold and p Mbe a point in M. Thetangent space TpMto Mat pis the real vector space of all tangent vectors to Mat p.

Let (U,x) be a local chart about p. Then the vectors

x1

p

, , xn

p

form a basis in the tangent space called the coordinate basis, or thecoor-

dinate frame.

Avector field Xon an open setU Min a manifold Mis the differentiableassignment of a tangent vectorXpto each point p

U.

In local coordinatesX =

nj=1

Xj(x)

xj.

Example.



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1.3.4 Mappings

LetMbe ann-dimensional manifold and Nbe anm-dimensional manifoldand letF : MNbe a map from Mto N. Let (U, )Abe an atlas in Mand (V, )Bbe an atlas inN. We define the maps

F = F 1 : (U)(V)

from open sets in Rn to Rm defined by

ya = Fa(x1, . . . ,xn) ,

wherea =1, . . . , m.

The map Fis said to be smoothifFa are smooth functions of local coor-dinatesxi,i =1, . . . , n.

The process of replacing the map Fby the functions F =

F

1 is

usually omitted, and the maps FandF are identified, so that we think ofthe mapFdirectly in terms of local coordinates.

Ifn = m and the map F : M N is bijective and both F and F1 aredifferentiable, thenFis called adiffeomorphism.

That is, a diffeomorphism is a differentiable homeomorphism with differen-tiable inverse.

If this is only true in a neighborhood of a point p M, then F is called alocal diffeomorphism.

Example.



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Definition 1.3.2 Let M and N be two manifolds and F : M N bea map from M into N. Let p0 M be a point in M and X Tp0M bea tangent vector to M at p0. Let p = p(t), t (, ), be a curve in Msuch that

p(0) = p0, p(0) = X.

Then thedifferentialof F is the map

F : Tp0MTF(p0)N

defined by

F

X =

d

dt

F(p(t))t=0 . Fdoes not depend on the curve p(t). The matrix of the linear transformationFin therms of the coordinate bases

/y and/xi is theJacobian matrix

(F)

i = y

xi ,

that is,

(FX)i =

n

i=1 y

xiXi .

1.3.5 Submanifolds

Definition 1.3.3 Let M be an n-dimensional manifold and W M bea subset of M. Then W is an r-dimensional embedded submanifoldof

M if W is locally described as the common locus of(nr)differentiableindependent functions

F(x1, . . . ,xn) =0 , =1, . . . , n r,

such that the Jacobian matrix has rank(n r)at each point of the locus,that is,

rank

F

xi (x)

=n r, xW.

Every embedded submanifold of a manifold is itself a manifold.



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Theorem 1.3.2 Let n and r be two positive integers such that n > r.

Let M be an n-dimensional manifold and N be an r-dimensional man-ifold. Let q N be a point in N such that the inverse image W =F1(q) is nonempty. Suppose that for each point p W the differ-ential F of the map F is surjective, that is, has the maximal rank

rankF(p) =r.

Then W is an (n r)dimensional submanifold of M.

Example. Morse Map. (Height function F : M Rfor trousers surfaceMin R3).

Letp0 Mand vTp0Mbe a tangent vector toMat p0. ThenF :Tp0MR =TF(p0)Mis the projection ofv to z-axis defined byF(v1, v2, v3) =v3.

Let p0 M and z = F(p0). F1(z) is an embedded submanifold ifF isonto, that is, Tp0M is not horizontal. IfTp0Mis horizontal, then F = 0(hence, not onto).

Definition 1.3.4 Let M and N be two manifolds and F : MN be adifferentiable map from M into N.

A point pM is aregular pointif the differential F : TpMTF(p)Nis surjective.

A point pM is acritical pointif it is not regular.

A point q N is a regular value of F if the inverse image F1(q) iseither empty or consists only of regular points.

A point qN is acritical valueof F if it is not regular.

Thus, we can reformulate the main theorem as follows.

Theorem 1.3.3 Let n and r be two positive integers such that n >

r. Let M be an n-dimensional manifold and N be an r-dimensional

manifold. Let q N be a regular value of F. Then the inverse imageW = F1(q) is either empty or is an(nr)dimensional submanifoldof M.



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1.4. VECTOR FIELDS AND FLOWS 21

1.4 Vector Fields and Flows

1.4.1 Vector Fields in Rn

Letx =(xj)Rn be a point in Rn andv(x)TxRn be a vector at x given by

v =

nj=1

vj(x)j,

wherej = /xj and the components vj(x) are smooth (or just differen-

tiable) functions ofx. Thenv(x) is called avector fieldin Rn.

Lett(, ) andt :R

n Rn

be a family of diffeomorphisms such that

0 =Id

and for anyt, t1, t2(, ) such that t, t1 + t2(, ) there holds

t1 t2 = t1+t2and

t =1

t .

Such aone-parameter groupof diffeomorphisms is called aflowon Rn.

A flowtdefines a vector fieldv by

v(x) = d

dtt(x)

t=0

with the components

vj(x) = dxj

dt .

The corresponding differential operator

v(f)(x) = d

dtf(t(x))

t=0

=

nj=1

vj(x)f

xj

is the derivative along the streamline of the flow through the point p.



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Conversely, every vector field v determines a flow, which is determined byrequiringv to be the velocity field of the flow.

Such a flow is defined as the solution of the system of ordinary first-orderdifferential equations

dxj

dt =vj(x1(t), . . . ,xn(t)) , j =1, . . . , n,

with the initial conditions

xj(0) = xj

0.

The solution of this system defines theintegral curvesof the vector fieldv.

Theorem 1.4.1 Fundamental Theorem on Vector Fields inRn LetRnbe an open set in Rn andv : U Rn be a smooth vector field on U.Then for any point x0 U there is > 0 and a neighborhood V of x0such that:

1. there is a unique curve (called the integral curve ofv) x(t) =

t(x0), t(, )such that for any t(, )

x(t) dx(t)dt

=v(x(t))

and x(0) = x0;

2. the map

V (, )Rn

defined by (x, t) t(x) is smooth and for any t1, t2 (, )such that t1 + t2(, )

t1 t2 =t1+t2holds in V. The family of maps tis called a local one-parameter

group of diffeomorphisms or alocal flow.

Remarks. The local flow is only defined in a small neighborhood of the pointx0. The one-parameter group is not a group in the strict sense.



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1.4. VECTOR FIELDS AND FLOWS 23

The integral curves exist only for a small time. If the vector field is not differentiable, then the integral curve is not unique.

1.4.2 Vector Fields on Manifolds

LetWbe an open subset of a mnifold Mand v be a smooth vector field onW.

Let (U,x) be a local chart in W. IfW U, then one can proceed as in Rn.

IfW is not contained in a single chart, then we choose a cover ofW andproceed as follows. Let p W and (U,x) and (U,x) be two chartscovering p.

Then the integral curves in both local coordinate systems have the samemeaning and define a unique integral curve in M. This defines a local flow

onW in M. We just need to check that if the flow equations are satisfied in

one coordinate system, then they are satisfied in another coordinate system.

1.4.3 Straightening Flows

LetUbe an open set in a manifold Mand t : U Mbe a local flow ona Msuch that0(p) = p. Then t(p) depends smoothly on both the time t

and the point p.

Note that if a vector field does not vanish at a point p, i.e. vp 0, then itdoes not vanish in a neighborhood ofp.

Let p0 be a fixed point in M and W be a sufficiently small hypersurfacepassing throughp0such thatvpistransversaltoWat every point ofpW.This just means thatv is nowhere tangent toW.

IfW is small enough it can be covered by a single coordinate chart. Let(u1, . . . , un1) be local coordinates forWsuch that the the point p0has coor-dinates (0, . . . , 0).

Then for a small neighborhood of a point p and sufficiently small t then-tuple (u1, . . . , un1, t) gives a local coordinate system near pin M.



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32/219

Chapter 2

Tensors

2.1 Covectors and Riemannian Metric

2.1.1 Linear Functionals and Dual Space

LetEbe a realn-dimensional vector space.

LetB ={e1, . . . , en} be a basis inE.

Then for anyv

E

v =n

j=1

vjej

The real numbers (v1, . . . , vn) are called the components ofv with respectto the basis B.

Remark. Every real vector space of dimension n is isomorphic to Rn.

Definition 2.1.1 Let E be a real vector space. Alinear functionalon

E is a linear real-valued function on E. That is, it is a map

: ER

satisfying the linearity conditions:a, b, R andv, wE

(av + bw) =a(v) + b(w) .

25


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26 CHAPTER 2. TENSORS

Given a basis {ej}, j =1, . . . , n, we defineaj =(ej) .

Then for anyvEwe have

(v) =

nj=1

ajvj .

Definition 2.1.2 Let E be a real vector space. The set of all linearfunctionals on E is called the dual spaceand denoted by E.

The dual space is a real vector space under the natural operations of addition

and multiplication by scalar defined by:, E,c1, c2R,vE,(c1 + c2)(v) =c1(v) + c2(v) .

Let{ej} be a basis in E and{j}, j = 1, . . . , n, be linear functionals on Esuch that

j(ei) =j

i.

Thenj(v) = vj .

Theorem 2.1.1 Let E be a real vector space and{ej}, j =1, . . . , n, bea basis in E. Let and{j}, j =1, . . . , n, be linear functionals on E suchthat

j(ei) =j

i.

Then the linear functionals {j} for a basis in the dual space E, calledthedual basisto the basis {ej}, so that for any E

=

nj=1

ajj .

The real numbersaj =(ej)

are called thecomponentsof the linear functionalwith respect to the

basis {j}.Proof:



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2.1. COVECTORS AND RIEMANNIAN METRIC 27

1.

2.1.2 Differential of a Function

Definition 2.1.3 Let M be a manifold and pM be a point in M. Thespace TpM dual to the tangent space TpM at p is called the cotangentspace.

Definition 2.1.4 Let M be a manifold and f : MR be a real valuedsmooth function on M. Let p M be a point in M. The differentiald f TpM of f at p is the linear functional

d f :TpR

defined by

d f(v) =vp(f) .

In local coordinates xj the differential is defined by

d f(v) =

nj=1

vj(x)f

xj

In particular,

d f

xj

=

f

xj.

Thus

dxi

xj

=ij.

and

dxi(v) = vi .

The differentials

{dxi

}form a basis for the cotangent space TpMcalled the

coordinate basis.

Therefore, every linear functional has the form

=

nj=1

ajdxj .



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That is why, the linear functionals are also called differential forms, or1-forms, orcovectors, or covariant vectors.

Definition 2.1.5 A covector field is a differentiable assignment of

a covectorp TpM to each point p of a manifold. This means thatthe components of the covector field are differentiable functions of local

coordinates.

Therefore, a covector field has the form

=

nj=1

aj(x)dxj

Under a change of local coordinates xj = xj(x) the differentials transformaccording to

dxj =

ni=1

xj

xidxi.

Therefore, the components of a covector transform as

ai =

nj=1

xj

xia

j.

2.1.3 Inner Product

LetEbe an-dimensional real vector space. Theinner product(or scalar product) on E is a symmetric bilinear non-

degenerate functional on EE, that is, it is a map, : EE Rsuchthat

1.v, wE,v, w =w, v

2.v, w, uE, a, bRav + bu, w = av, w + bu, w

3.vE,if v, w =0, wE, thenv =0 .



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4. If, in addition, vE,v, v 0

and

v, v =0 if and only if v =0 ,then the inner product is called positive definite.

For a positive-definite inner product the normof a vectorv is defined by

||v|| =

v, v

Let

{ej

}be a basis inE.

Then the matrixgi jdefined by

gi j =ei, ej

is a metric tensor, more precisely it gives the components of the metric

tensor in that basis.

The matrixgi jis symmetric and nondegenerate, that is,

gi j =gji, det gi j 0 .

For a positive definite inner product, this matrix is positive-definite, that

is, it has only positive real eigenvalues. One says, that the metric has the

signature (+ +). In special relativity one considers metrics wich are notpositive definite but have the signature ( + +).

Two vectorsv, wEare orthogonalif

v, w =0 .

A vectoruEis calledunit vectorif

||u|| =1 .

The basis is called orthonormalif

gi j =i j.



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The inner product is given then by

v, w =n

i,j=1

gi jviwj .

LetvE. Then we can define a linear functionalEby

(w) =v, w .

Therefore, each vectorvEdefines a covectorEcalled thecovariantversion of the vector v.

Given a basis {ej} inEand the dual basisj inE we find

i =

nj=1

gi jvj .

This operation is calledlowering the indexof a vector. Therefore, we can denote the components of the covector corresponding

to a vectorv by the same symbol and call them thecovariant components

of the vector.

In an orthonormal basis, of course,

vi =vi .

Similarly, given a covector E we can define a vector v Esuch thatwE

(w) =v, w .

Letgi j represent the matrix inverse to the matrix gi j.

Then the contravariant components can be computed from the covariantcomponents by

vi =

nj=1

gi jvj.

This operation is calledraising the index of a covector.



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Thus, the vector spaces Eand E are isomorphic. The isomorphism is pro-vided by the inner product (the metric).

The spaceEcan also be considered as the space of linear functionals on E.A vectorvEdefines a linear functional v : ER by, for anyE

v() =(v).

2.1.4 Riemannian Manifolds

Definition 2.1.6 Let M be a manifold. ARiemannian metricon M is

a differentiable assignment of a positive definite inner product in eachtangent space TpM to the manifold at each point pM.

If the inner product is non-degenerate but not positive definite, then it is

apseudo-Riemannian metric.

ARiemannian manifoldis a pair(M, g)of a manifold with a Rieman-

nian metric on it.

LetpMbe a point in Mand (U,x) be a local coordinate system about p.Letibe the coordinate basis in TpMand g

i j

=i,

jbe the components

of the metric tensor in the coordinate system x. Let (U,x) be anothercoordinate system containing p. Then the components of the metric tensor

transform as

gi j =

nk,l=1

xk

xi

xl

xj

g

kl.

Definition 2.1.7 Let(M, g)be a Riemannian manifold and f : MRbe a smooth function. Then thegradient of f is a vector field grad f

associated with the covector field d f . That is, for any p M and anyv

TpM

grad f, v =d f(v) .

In local coordinates,( grad f)i =

nj=1

gi jf

xj.



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2.1.5 Curves of Steepest Ascent

Let (M, g) be a Riemannian manifold. Let p Mbe a point in M. For positive-definite inner product there is the

Schwartz inequality: for anyv, wTpM,|v, w| ||v|| ||w|| .

LetuTpMbe a unit vector. Then for any f C(M)u(f) =grad f, u .

Therefore,

|u(f)| =|grad f, u| || grad f|| . Thus, fhas a maximum rate of change in the direction of the gradient.

Definition 2.1.8 Let a R be a real number. Then the level set off : MR is the subset of M defined by

S = f1(a) ={pM| f(p) =a} .

Let f : MR be a smooth function,aR andS = f1(a) be the level setof f. Let p0

Sbe a point in the level setSand let p = p(t),t

(

, ), be

a curve inSso that p(0) = p0, f(p(t)) =a and p(0)Tp0 S. Thend

dtf(p(t)) =grad f, p =d f( p) =0 .

Thus, the gradient of fis orthogonal to the level set of f. The flow

d p

dt =

grad f

|| grad f||defined by the gradient of a smooth function f is called the Morse defor-

mation. It has the property that

d f

dt=1

and, therefore, in timetit maps the level set f1(a) into the level set f1(a +t).



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2.2. TANGENT BUNDLE 33

2.2 Tangent Bundle

2.2.1 Fiber Bundles

LetMand Ebe smooth manifolds and : E Mbe a smooth map. Thenthe triple (E, ,M) is called abundle.

The manifoldMis called thebase manifoldof the bundle and the manifoldEis called thebundle spaceof the bundle (or thebundle space manifold).

The map is called theprojection.

The inverse image1(p) of a point pMis called thefiberover p.

The projection map is supposed to be surjective, that is, the differentialhas the maximal rank equal to dimM.

Let {U}Abe an atlas of local charts covering the base manifold Mand letU =U U,U =U U Uetc.

Afiber bundleis a bundle all fibers of which, 1(p),p M, are diffeo-morphic to a common manifoldFcalled thetypical fiberof the bundle (or

just the fiber).

For a fiber bundle, the inverse images1(U) are diffeomorphic toU F.That is, there are diffeomorhisms

h :U F1(U) ,such that for any pU M,F

(h(p, )) = p .

The diffeomorphisms

=h1

h :U FU Fare called thetransition functionsof the bundle.

The transition functions are defined by, pU M, F,(p, ) =(p, (R(p))()) .

That is, for all pU there are diffeomorphismsR(p) of the fiberR(p) : F F.



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It is required that the set of all transformationsR(p) Gfor all, andpU Mforms a groupG. This group is called thestructure groupofthe bundle.

Thus, the transition functionsdetermine smooth maps

R :U G ,

that assign to each point p U an element R(p) G of the structuregroup.

Of course, from the definition of these maps we immediately obtain theconsistency conditions(orcompatibility conditions)

R(p) =(R(p))1 , pU,

R(p)R(p)R(p) =IdM, pU.

Aprincipal bundleis a fiber bundle (E, ,M) whose fiberFcoincides withthe structure group, that is, F = G.

A fiber bundle with any fiber Fis fully determined by the transition func-tions satisfying the consistency conditions. The fiber Fdoes not play much

role in this construction.

LetFbe a manifold and Diff(F) be the set of all diffeomorphisms F F.Let G be a group and e G be the identity element ofG. Then a mapT :GDiff(F) such that

T(e) = IdF,

T(R1) = (T(R))1 , R G ,T(R1R2) = T(R1) T(R2) , R1,R2 G ,

is called arepresentationof the groupG.

Given a fiber bundle (E, ,M) with a fiber Fand a structure group G onecan construct another fiber bundle (E, ,M) with a fiber F and the samestructure group G as follows. One takes a representation of the structure

groupT : G Diff(F) on the fiber F and simply replaces the transitionfunctionsR byT(R). Such a fiber bundle is called a bundle associated

with the original bundle.



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Thus, every fiber bundle is an associated bundle with some principal bun-dle. So, all bundles can be constructed as associated bundles from principalbundles. All we need is the structure group. The fiber is not important.

Avector bundleis a fiber bundle whose fiber is a vector space.

Asectionof a bundle (E, ,M) is a map s : M Esuch that the image ofeach point pMis in the fiber1(p) over this point, that is, s(p)1(p),or

s =IdM.

2.2.2 Tangent Bundle

Definition 2.2.1 Let M be a smooth manifold. Thetangent bundle

T M to M is the collection of all tangent vectors at all points of M.

T M ={(p, v)|pM, vTpM}

Let dimM =n.

Let p Mbe a point in the manifold M, (U,x) be a local chart and (xi) bethe local coordinates of the point p.

Leti =/xi be the coordinate basis forTpM.

Letv = ni=1vii TpM. Then the local coordinates of the point (p, v)T Mare

(x1, . . . ,xn, v1, . . . , vn) .

Remarks.

The coordinates (xi) are local; they are restricted to the local chartU, thatis, (xi)U Rn.

The coordinates vi are not restricted, that is, (vi)Rn, they take any valuesin Rn.

The open setURn R2n is a local chart in the tangent bundle T M.

Let (U,x) and (U,x) be two local charts containing the point p.



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Then the local coordinates of the point (p, v) in overlapping local charts arerelated by

xi = xi(x)

vi =

nj=1

xi

xj

vj

This is a local diffeomorphism. Thus, the tangent bundle T Mis a manifold of dimension 2 dimM.

A map : T M

Mdefined by

(p, v) = p

is called theprojection map. It assigns to a vector tangent to M the point

in Mat which the vector sits.

Locally, ifphas coordinates (x1, . . . ,xn) andv has components (v1, . . . , vn)in the coordinate basis, then

(x1, . . . ,xn, v1, . . . , vn) =(x1, . . . ,xn) .

Let p Mbe a point in M. The set1(p) T Mis called thefiberof thetangent bundle.

The fiber of the tangent bundle

1(p) =TpM

is the tangent space at p.

Remarks.

There is no global projection map :T M

Rn defined by(p, v) =v.

In general,T M M Rn. For any chartU M

1(U) =URn .Thus, locally the tangent bundle is a product manifold.



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A vector field is a mapv: MT M,

such that

v =Id : M M.

A vector field is a cross sectionof the tangent bundle. The image of the manifold under a vector field is a n-dimensional subman-

ifold of the tangent bundleT M.

The zero vector field defines thezero sectionof the tangent bundle.

Definition 2.2.2 Let (M, g) be an n-dimensional Riemannian mani-fold. Theunite tangent bundle of M is the set T0M of all unit vectors

to M,

T0M ={(p, v)|pM, vTpM, ||v|| =1} ,where, locally, ||v||2 = ni,j=1gi j(p)vivj.

The unit tangent bundle is a (2n1)-dimensional submanifold of the tangentbundleT M.

Theorem 2.2.1 Let S2 be the unit2-sphere embedded in R3. The unit

tangent bundle T0S2

is homeomorphic to the real projective space RP3

and to the special orthogonal group S O(3)

T0S2 RP3 S O(3) .

Proof:

1.



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2.3 The Cotangent Bundle

Definition 2.3.1 Let M be a smooth manifold. Thecotangent bundle

TM to M is the collection of all covectors at all points of M

TM ={(p, )| pM, TpM}

Let dimM =n. Let p Mbe a point in the manifold M, (U,x) be a local chart and (xi) be

the local coordinates of the point p.

Letd xi be the coordinate basis forTpM.

Let = ni=1idxi TpM. Then the local coordinates of the point (p, )TMare

(x1, . . . ,xn, 1, . . . , n) .

Remarks. The open setURn R2n is a local chart in the cotangent bundle TM. Let (U,x) and (U,x) be two local charts containing the point p.

Then the local coordinates of the point (p, ) in overlapping local charts are

related by

xi = xi(x)

i =

nj=1

xj

xi

j.

This is a local diffeomorphism. Thus, the cotangent bundleTMis a manifold of dimension 2n.

The projection map : TM

Mis defined by(p, ) = p.

A covector field (or a 1-form)is a map: MTM,

such that

=Id : M M.



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2.3. THE COTANGENT BUNDLE 39

A covector field is asectionof the cotangent bundle.

2.3.1 Pull-Back of a Covector

LetMand Nbe two smooth manifolds andn =dimMand m =dimN. Let : MNbe a smooth map. The differential

:TpMT(p)Nis the linear transformation of the tangent spaces.

Letxi

be a local coordinate system in a local chart about pMand y

be alocal coordinate system in a local chart about(p)Nand iandbe thecoordinate bases forTpMand T(p)N.

Then the action of the differential is defined by

xj

=

m=1

y

xj

y .

Letv =

ni=1v

ii. Then

[(v)]

=

nj=1

y

xjvj .

Definition 2.3.2 Thepullback is the linear transformation of thecotangent spaces

: T(p)NTpMtaking covectors at(p)N to covectors at pM, defined as follows.IfT

(p)N, then()TpM so that

() =

: TpMRwhere : T(p)N R. That is, for any vectorvTpM

(()) (v) =((v)) .

Diagram.



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In local coordinates,

[(dy)]j =m

=1

y

xjdxj .

Let = m=1dy. Then() =

nj=1

m=1

y

xjdxj ,

that is, in components,

[()]j =

m

=1

y

xj .

Remark. In general, for a map : M N, the following linear transformations

are well defined: the differential : TpM T(p)N and the pullback :T(p)NTpM.

The maps TpM T(p)N and T(p)N TpM are not well defined, ingeneral.

If dimM =dimNand : MNis a diffeomorphism, then all these mapsare well defined.

Explain.

2.3.2 Phase Space

LetMbe aconfiguration spaceof a dynamical system with localgeneral-ized coordinatesq1, . . . , qn. Then qi =

dqi

dt are thegeneralized velocities.

Under a change of local coordinatesqi =q

i(q)

the velocities transform as components of a vector

qj =

ni=1

qj

qiqi.



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Therefore, (q1, . . . , qn,q1, . . . ,qn) give local coordinates for the tangent bun-dleT M.

LetL : T MR be a map. ThenL(q,q) is called theLagrangian.

Thegeneralized momenta piare defined by

pi = L

qi.

The momenta are functions on T M, that is, p: T MR.

Under a change of local coordinatesq = q(q) the momenta transform ascomponents of a covector

pj =

ni=1

qi

qj

p

i ,

The matrixHik(q,q) =

2L

qiqk

is called theHessian.

Suppose that the Hessian is non-degerate

det Hik0 .

Then the velocities can be expressed in terms of momenta

qi = qi(q,p) ,

that is, there is a map q : TM

R. More generally, there is a map

T MTM.

Thus, (q1, . . . , qn,p1, . . . ,pn) give local coordinates for the cotangent bundleTM.

The cotangent bundle is called thephase spacein dynamics.



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TheHamiltonianis a smooth function on the cotangent bundleH :TMR defined by

H(q,p) =

ni=1

L

qiqi L(q, q) .

Example.

One of the most important examples is the Lagrangian quadratic in veloci-ties

L(q,q) = 1

2

ni,j=1

gi j(q)qiqj V(q) ,

wheregi jis a Riemannian metric on Mand Vis a smooth function on M.

Then the Hessian isgik =

2L

qiqk

and, therefore, nondegenerate.

The relation between momenta and the velocities is

pi =

n

j=1gi j(q)q

j , qi =

n

j=1gi j(q)pj.

The Hamiltonian is given by

H(q,p) = 1

2

ni,j=1

gi j(q)pipj + V(q) ,

2.3.3 The Poincare1-Form

ThePoincare1-form is a 1-form on the cotangent bundle TM definedin local coordinates (q,p) onTMby

=

ni=1

pidqi .

Remarks.



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The coordinates pare not functions on M. The Poincare form is not a 1-form on M. A general 1-form onTMis

=

ni=1

i(q,p)dqi

+

ni=1

vi(q,p)d pi.

Theorem 2.3.1 The Poincare 1-form is well defined globally on thecotangent bundle of any manifold.

Proof:

1. Let (q,p) and (q,p) be two overlapping coordinate patches ofTM.

2. Then

dqi =

nj=1

qi

qj

dqj

andn

i=1

pidqi =

nj=1

p

j dqj

.

We give now an intrinsic definition of the Poincare form. Let (q,p) TM be a point in TM. We want to define a 1-form

T(q,p)(TM) at this point (q,p)TM.

Let : TM Mbe the projection defined for any q M,p TqMby(q,p) =q.

Then the pullback is the map : TqM T(q,p)(TM). For each 1-formp TqM it defines a 1-form (p) T(q,p)(TM). This is precisely thePoincare 1-form, that is,

=(p) . Of course, in local coordinates

(p) =n

i=1

pidqi .



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2.4 Tensors

2.4.1 Covariant Tensors

LetEbe a vector space andEbe its dual space.

Leteibe a basis for Eand i be the dual basis for E.

Acovariant tensor of rank p(or a tensor of type(0,p)) is a multi-linearreal-valued functional

Q: E E

p

R

Remarks.

The functionQ(v1, . . . , vp) is linear in each argument.

The functional Q is independent of any basis.

A covariant vector (covector) is a covariant tensor of rank 1.

A metric tensor is a covariant tensor of rank 2.

Thecomponents of the tensor Qwith respect to the basis ei are definedby

Qi1...ip = Q(ei1 , . . . , eip ) .

Then for any vectorsva =

nj=1

vjaej,

wherea =1, . . . ,p, we have

Q(v1, . . . , vp) =

n

j1,...,jp=1Qj1...jp v

j11 vjpp .

The collection of all covariant tensors of rank p forms a vector space de-noted by

Tp = E E

p

.



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2.4. TENSORS 45

The dimension of the vector spaceTpis

dim Tp =np .

Thetensor productof two covectors, Eis a covariant tensor E Eof rank 2 defined by:v, wE

( )(v, w) =(v)(w) .

The components of the tensor product are

(

)i j

=i

j.

Thetensor product of a covariant tensor Qof rank pand a covariant ten-sor T of rank q is a covariant tensor QT of rank (p + q) defined by:v1, . . . , vp, w1, . . . , wqE

(Q T)(v1, . . . , vp, w1, . . . , wq) = Q(v1, . . . , vp)T(w1, . . . , wq) .

The components of the tensor productQ T are

(Q

T)i1...ipj1...jq = Qi1...ip Tj1...jq.

Thus,: Tp TqTp+q.

Tensor product is associative. The basis in the spaceTpis

i1 ip ,

where 1i1, . . . , ipn. A covariant tensor Q of rankphas the form

Q =

ni1,...,ip=1

Qi1...ip i1 ip .



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2.4.2 Contravariant Tensors

A contravariant vector can be considered as a linear real-valued functional

v: ER .

Acontravariant tensor of rank p (or a tensor of type (p, 0)) is a multi-linear real-valued functional

T :E E

p

R

Remarks.

The functionT(1, , p) is linear in each argument.

The functionalTis independent of any basis.

A contravariant vector (covector) is a contravariant tensor of rank 1.

Thecomponents of the tensor Twith respect to the basis i are definedby

Ti1...ip =T(i1 , . . . , ip ) .

Then for any covectors(a) =

nj=1

(a) j j ,

wherea =1, . . . ,p, we have

T((1), . . . , (p)) =

nj1,...,jp=1

Tj1...jp (1) j1 (p) jp.

The inverse matrix of the components of a metric tensor defines a con-travariant tensorg1 of rank 2 by

g1(, ) =n

i,j=1

gi jij.



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2.4. TENSORS 47

The collection of all contravariant tensors of rank p forms a vector spacedenoted by

Tp = E Ep

.

The dimension of the vector spaceTp is

dim Tp =np .

Thetensor productof a contravariant tensorQ of rankpand a contravari-ant tensorTof rankqis a contravariant tensorQ Tof rank (p + q) definedby:

1, . . . ,

p,

1, . . . ,

qE

(Q T)(1, . . . , p, 1, . . . , q) = Q(1, . . . , p)T(1, . . . , q) .


(Q T)i1...ipj1...jq = Qi1...ip Tj1...jq .

Thus,: Tp Tq Tp+q .

Tensor product is associative.

The basis in the spaceTp is

ei1 eip,

where 1i1, . . . , ipn.

A contravariant tensorTof rankphas the form

T =

ni1,...,ip=1

Ti1...ip ei1 eip.

The set of all tensors of type (p, 0) forms a vector spaceTp of dimensionnp

dim Tp =np .



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2.4.3 General Tensors of Type(p, q)

Atensor of type(p, q) is a multi-linear real-valued functional

T : E Ep

E Eq

R

The components of the tensor Twith respect to the basis ei, i are definedby

Ti1...ip

j1...jq=T(i1 , . . . , ip , ej1 , . . . , ejq ) .

Then for any covectors(a) =

nj=1

(a) j j ,

wherea =1, . . . ,p, and any vectors

vb =

nj=1

vkbek,

whereb =1, . . . , q, we have

T((1), . . . , (p), v1, . . . , vq) =

n

k1,...,kq=1n

j1,...,jp=1 Tj1...jpk1...kq (1) j1 (p) jp vk1 vkq . The inverse matrix of the components of a metric tensor defines a con-

travariant tensorg1 of rank 2 by

g1(, ) =n

i,j=1

gi jij.

The collection of all tensors of type (p, q) forms a vector space denoted by

Tpq = E E

p

E E

q

.

The dimension of the vector space Tpis

dim Tpq =np+q .



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2.4. TENSORS 49

Thetensor productof a tensorQ of type (p, q) and a tensorTof type (r,s)is a tensorQTof type (p+r, q+ s) defined by:1, . . . , p, 1, . . . , r E,v1, . . . , vq, w1, . . . , wsE

(Q T)(1, . . . , p, 1, . . . , r, v1, . . . , vq, w1, . . . , ws)= Q(1, . . . , p, v1, . . . , vq)T(1, . . . , r, w1, . . . , ws) .


(Q T)i1...ipj1...jrk1...kql1...ls

= Qi1...ip

k1...kqT

j1...jrl1...ls

.

Thus,: Tpq Trs Tp+rq+s .

We stress once again that the tensor product is associative.

The basis in the spaceTpq is

ei1 eip j1 jq ,

where 1i1, . . . , ip, j1, . . . , jqn.

A tensorTof type (p, q) has the form

T =

nj1,...,jq=1

ni1,...,ip=1

Ti1...ip

j1...jqei1 eip j1 jq .

Let p, q1 and 1r p, 1 sq. The (r,s)-contractionof tensors oftype (p, q) is the map

tr rs : Tpq Tp1q1

defined by

(tr rs T)i1...ip1j1...jq1

=

nk=1

Ti1...ir1kir...ip1j1...js1k js...jq

.



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2.4.4 Linear Transformations and Tensors

LetA : EEbe a linear transformation. Then we can define a tensor Aof type (1, 1) by E, vE

A(, v) =(Av) .

Let (Aij) be the matrix of the linear transformationA, that is,

Aej =

ni=1

Aijei.

Then the components of the tensor AareAij = A(

i, ej) =i(Aej) = A

ij.

Thus, one can identify tensors of type (1, 1) and linear transformations onE(and, similarly on E as well).

Then,A =

ni,j=1

Aijei j .

The identity linear transformationIhas the matrixIij =

ij

and defines the tensor of type (1, 1)

I =

ni=1

ei i .

To summarize, a (1, 1) tensor A : EE R is identified with the lineartransformationA : E

E.

The covariant tensor Ai j, the contravariant tensor Ai j and the tensor Aij oftype (1, 1) are related by

Ai j =

nk=1

gikAk

j, Ai j

=

nk=1

Aikgk j .



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2.4. TENSORS 51

2.4.5 Tensor Fields

Definition 2.4.1 A tensor field on a manifold M is a smooth assign-ment of a tensor at each point of M.

Letxi = xi(x) be a local diffeomorphism.

Thendxi =

nj=1

xi

xj

dxj

and

xi

=

nj=1

x

j

xi

xj

LetTbe a tensor of type (p, q). Then

T()i1...ip

j1...jq=

nk1,...,kp=1

nl1,...,lq=1

xi1

xk1

xip

xkp

xl1

xlq

x

j1

xjq

T()k1...kp

l1...lq

2.4.6 Tensor Bundles

Definition 2.4.2 Let M be a smooth manifold. Thetensor bundle of

type(p, q)TpqM is the collection of all tensors of type(p, q)at all points

of M

TpqM ={(p, T)|pM, T Tpq, (x)M}

The tensor bundle TpqMis the tensor product of the tangent and cotangentbundles

TpqM =T M T M

p

TM TM

q

.

Let dimM =n.

Let x Mbe a point in the manifold M, (U,x) be a local chart and (xi) bethe local coordinates of the point p.

Letiand d xi be the coordinate basis forTxMand TxM.



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LetTbe a tensor of type (p, q) andTi1...ipj1...jq be the components of the tensorTin the coordinate basis. Then the local coordinates of the point (p, T)T

pqM

are

(xi, Ti1...ip

j1...jq) ,

where 1i, i1, . . . , ip, j1, . . . , jqn.

Remarks.

The open setURnp+q Rn+np+q is a local chart in the tensor bundle TpqM.

The bundleTpqMis a manifold of dimension n + np+q.

The projection map : TpqM Mis defined by(x, T) = x.

Atensor fieldof type (p, q) is a map

T : MTpqM,

such that

T =Id : M M.

A tensor field of type (p, q) is asectionof the tensor bundle TpqM.

2.4.7 Examples

Riemannian metricg.

Energy-momentum tensorT.

Stress tensori j.

Riemann curvature tensorR.

Ricci curvature tensor R. Scalar density of weight 1:|g| = det gi j. Axial vectors (vector product) in R3.

Strength of the electromagnetic field F.



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2.4. TENSORS 53

Theorem 2.4.1 Let

A =

ni=1

Aidxi

be a covector field (1-form). Let

Fi j =iAj jAi.

Then

F =

ni,j=1

Fi jdxi dxj

is a tensor of type (0, 2).

Proof:

1. Check the transformation law.

A tensor is calledisotropicif it is a tensor product ofg,g1 andI.

The components of an isotropic tensor are the products ofgi j,gi j andij.

Every isotropic tensor of type (p, q) has an even rankp + q.

For example, the most general isotropic tensor of type (2, 2) has the form

Ai jkl =agi jgkl + b

ik

j

l + cil

j

k,

wherea, b, care scalars.

2.4.8 Einstein Summation Convention

In any expression there are two types of indices: free indicesandrepeated

indices.

Free indices appear only once in an expression; they are assumed to take allpossible values from 1 to n.

The position of all free indices in all terms in an equation must be the same.



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Repeated indices appear twice in an expression. It is assumed that there is asummation over each repeated pair of indices from 1 to n. The summationover a pair of repeated indices in an expression is called thecontraction.

Repeated indices are dummy indices: they can be replaced by any otherletter (not already used in the expression) without changing the meaning of

the expression.

Indices cannot be repeated on the same level. That is, in a pair of repeatedindices one index is in upper position and another is in the lower position.

There cannot be indices occuring three or more times in any expression.



62/219

Chapter 3

Differential Forms

3.1 Exterior Algebra

3.1.1 Permutation Group

Agroupis a setGwith an associative binary operation, : G G Gwithidentity, called the multiplication, such that each element has an inverse.

That is, the following conditions are satisfied

1. for any three elementsg, h, k

G, theassociativity lawholds: (gh)k =

g(hk);

2. there exists anidentity elemente G such that for any g G,ge =eg =g;

3. each elementg Ghas aninverseg1, such thatg g1 =g1 g =e

LetXbe a set. A transformationof the setXis a bijective mapg : XX.

The set of all transformations of a set Xforms a group Aut(X), with com-position of maps as group multiplication.

Any subgroup of Aut(X) is atransformation groupof the set X.

The transformations of a finite set Xare calledpermutations.

The groupSp of permutations of the set Zn ={1, . . . ,p}is called the sym-metric group of order p.

55


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56 CHAPTER 3. DIFFERENTIAL FORMS

Theorem 3.1.1 The order of the symmetric group Sp is|Sp| = p! .

Any subgroup ofSp is called apermutation group. A permutation : ZpZpcan be represented by

1 . . . p

(1) . . . (p)

The identity permutation is

1 . . . p1 . . . p The inverse permutation1 :ZpZpis represented by

(1) . . . (p)

1 . . . p

The product of permutations is then defined in an obvious manner. Anelementary permutation is a permutation that exchanges the order of

only two elements.

Every permutation can be realized as a product of elementary permutations. A permutation that can be realized by an even number of elementary per-

mutations is called aneven permutation.

A permutation that can be realized by an odd number of elementary permu-tations is called anodd permutation.

Proposition 3.1.1 The parity of a permutation does not depend on therepresentation of a permutation by a product of the elementary ones.

That is, each representation of an even permutation has even number of

elementary permutations, and similarly for odd permutations.

The sign of a permutation , denoted by sign() (or simply (1)), isdefined by

sign() =(1) =

+1, if is even,

1, if is odd



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3.1. EXTERIOR ALGEBRA 57

3.1.2 Permutations of Tensors

LetSp be the symmetric group of order p. Then every permutation Spdefines a map

: TpTp,which assigns to every tensor Tof type (0,p) a new tensor (T), called a

permutation of the tensorT, of type (0,p) by:v1, . . . , vp(T)(v1, . . . , vp) =T(v(1), . . . , v(p)) .

Let (i1, . . . , ip) be a p-tuple of integers. Then a permutation : Zp Zpdefines an action

(i1, . . . , ip) =(i(1), . . . , i(p)) .

The components of the tensor(T) are obtained by the action of the permu-tation on the indices of the tensorT

(T)i1...ip =Ti(1)...i(p).

Thesymmetrizationof the tensorTof the type (0,p) is defined by

Sym(T) = 1

p!

Sp

(T) .

The symmetrization is also denoted by parenthesis. The components of thesymmetrized tensor Sym(T) are given by

T(i1...ip) = 1

p!

Sp

Ti(1)...i(p).

Theanti-symmetrizationof the tensorTof the type (0,p) is defined by

Alt(T) = 1

p!

Sp

sign ()(T) .

The anti-symmetrization is also denoted by square brackets. The compo-nents of the anti-symmetrized tensor Alt(T) are given by

T[i1...ip] = 1

p!

Sp

sign ()Ti(1)...i(p).



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Examples. A tensorTof type (0,p) is calledsymmetricif for any permutation Sp

(T) =T.

A tensor Tof type (0,p) is called anti-symmetric if for any permutationSp

(T) =sign ()T.

An anti-symmetric tensor of type (0,p) is called a p-form.

Remarks. Permutation, symmetrization, anti-symmetrization of tensors of type (p, 0). Completely symmetric and completely anti-symmetric tensors of type (p, 0). An anti-symmetric tensor of type (p, 0) is called a p-vector. Partial permutation. Examples.

Notation.

3.1.3 Alternating Tensors

Let (i1, . . . , ip)a nd(j1, . . . , jp) be twop-tuples of integers 1i1, . . . , ip, j1, . . . , jpn. Thegeneralized Kronecker symbolis defined by

i1... ip

j1...jp=

1 if (i1, . . . , ip) is an even permutation of (j1, . . . , jp)

1 if (i1, . . . , ip) is an odd permutation of (j1, . . . , jp)0 otherwise

One can easily check that

i1... ip

j1...jp=det

i1j1

. . . i1jp

... . . .

...

ip

j1. . .

ip

jp



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Also, there holds

i1... ip

j1...jp = p!i1[j1

ip

jp].

Thus, the Kronecker symbolsi1... ipj1...jp are the components of the tensorsp!Alt(I I

p

)

of type (p,p), which are anti-symmetric separately in upper indices and the

lower indices.

Thus, the anti-symmetrization can also be written as

T[i1...ip] = 1

p! j

1...j

pi1...ip Tj1...jp.

Notation. Obviously, the Kronecker symbols vanish for p> n

i1...ip

j1...jp=0 ifp> n .

The contraction of Kronecker symbols gives Kronecker symbols with lowerindices, more precisely, we have the theorem.

Theorem 3.1.2 For any p, qN,1p, qn, there holds

i1...ipl1...lq

j1...jpl1...lq=

(n p)!(n q)!

i1...ip

j1...jp.

Proof:

1.

Corollary 3.1.1

For any qN,1

q

n we have

i1...iq

i1...iq=

n!

(n q)!.

In particular,

i1...ini1...in

=n! .



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Lemma 3.1.1 There holds

i1...ip j1 ...jr

l1...lpm1...mr

k1...krj1...jr

=r!i1...ipk1 ...kr

l1...lpm1...mr

Proof:

1.

In general, letA be a p-form (an antisymmetric tensor of type (0,p)) andBbe a p-vector (an anti-symmetric tensor of type (p, 0)). Then

Ai1...ip i1...ip

j1

...jp

= p!Aj1...jp,

Bi1...ip j1...jp

i1... ip= p!Bj1...jp .

Let (i1, . . . , in) be ann-tuple of integers 1 i1, . . . , in n. The completelyanti-symmetric (alternating)Levi-Civita symbolsare defined by

i1...in = 1 ... ni1...in

, i1...in =i1...in1 ... n

,

so that

i1...in =i1

...in

= 1 if (i1, . . . , in) is an even permutation of (1, . . . , n)

1 if (i1, . . . , in) is an odd permutation of (1, . . . , n)

0 otherwise

Theorem 3.1.3 There holds the identity

i1...in j1...jn =Sn

sign()i1j(1)

inj(n)

= n!i1[j1

injn]

= i1...inj1...jn

.

The contraction of this identity over k indices gives

i1...inkm1...mkj1...jnkm1...mk = k!(n k)!i1[j1 inkjnk]

= k!i1...inkj1...jnk

.

In particular,

m1...mn m1...mn =n! .



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It is easy to see that there holds also

i1...inpl1...lnp

j1...jpl1...lnp =(n p)!j1...jpi1...inp

The set of alln nreal matrices is denoted by Mat(n,R). Thedeterminantis a map det : Mat(n,R)R that assigns to each matrix

A =(Aij) a real number detAdefined by

detA =Sn

sign ()A1(1) An(n),

The most important properties of the determinant are listed below:

Theorem 3.1.4 1. The determinant of the product of matrices is equal to

the product of the determinants:

det(AB) =detA detB .

2. The determinants of a matrix A and of its transpose AT are equal:

detA =detAT .

3. The determinant of the inverse A1 of an invertible matrix A is equalto the inverse of the determinant of A:

detA1 =(detA)1

4. A matrix is invertible if and only if its determinant is non-zero.

The determinant of a matrix A =(Aij) can be written as

detA = i1...inA1i1. . .An

in

= j1...jnAj1

1. . .Ajn

n

= 1

n!i1...in j1...jnA

j1i1. . .A

jnin.

Here, as usual, a summation over all repeated indices is assumed from 1 to

n.



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3.1.4 Exteriorp-forms

An exterior p-form (or simply a p-form) is an anti-symmetric covarianttensorTpof type (0,p).

The collection of all p-forms forms a vector space p, which is a vectorsubspace ofTp

pTp.

In particular,0 = R and 1 =T1 = E

.

In other words, a 0-form is a smooth function, and a 1-form is a covector

field.

Let pbe a p-form. Leteibe a basis in Eand i be the dual basis in E. The components of the p-form are

i1...ip =(ei1 , . . . eip ) .

The components are completely anti-symmetric in all indices, that is,

i1...ip =sign ()i(1)...i(p).

In particular, under a permutation of any two indices the form changes sign

... i ... j ... =... j ... i ...,

which means that the components vanish if any two indices are equal

... i ... i ... =0 (no summation!) .

Thus, all non-vanishing components have different indices. Therefore, the values of all componentsi1...ip are completely determined by

the values of the components with the indices i1, . . . , ipreordered in strictly

increasing order

1i1


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Notation. To deal with forms it is convenient to introduce multi-indices. We will

denote a p-tuple of integers from 1 ton by a capital letter

I =(i1, . . . , ip) ,

where 1 i1, . . . , ip n. For a p-tuple of the same integers ordered in anincreasing order we define

I =(i1, . . . , ip) .

where 1i1


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There are no p-forms with p> n. Similarly to the norm of vectors and covectors we define theinner product

of exterior p-forms and in a Riemannian manifold by

(, ) = 1

p!gi1j1 gi1jp i1...ipj1...jp.

This enables one to define also the norm of an exterior p-form by|||| =

(, )

3.1.5 Exterior Product

Since the tensor product of two skew-symmetric tensors is not a skew-symmetric tensor to define the algebra of antisymmetric tensors we need to

define theanti-symmetric tensor productcalled the exterior(or wedge)

product.

If is an p-form and is anq-form then the exterior product of and isan (p + q)-form defined by

= (p + q)!p!q!

Alt ( ) .

In components( )i1...ip+q =

(p + q)!

p!q! [i1...ipip+1...ip+q].

Let pbe a p-form. Thenp =deg()

is called thedegree(orrank) of.

Theorem 3.1.6 The exterior product has the following properties

( ) = ( ) (associativity) =(1)deg()deg() (anticommutativity)( +) = + (distributivity) .

Proof:



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1.

The exterior square of any p-form of odd degree p(in particular, for any1-form) vanishes

=0 . Theexterior algebra (or Grassmann algebra) is the set of all forms of

all degrees, that is,

= 0 n. The dimension of the exterior algebra is

dim =

np=0

np =2n .

A basis of the space pisi1 ip , (1i1


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1.

Theorem 3.1.8 Letj 1 = E, 1 j p, be a collections of p1-forms andviE, 1ip, be a collection of p vectors. Let

Aji =j(vi) , 1i, jp .

Then 1 p

(v1, . . . , vp) =detA

ij.

Proof:

1.

Theorem 3.1.9 A collections of p 1-formsj 1 = E, 1 j p,is linearly dependent if and only if

1 p =0 .

Corollary 3.1.2 Let xi = xi(x), i = 1, . . . , n, be a local diffeomor-phism. Then

dx1 dxn =det xlxm

dx1 dxn .

3.1.6 Interior Product

Theinterior product of a vector v and a p-form is a (p1)-form ivdefined by, for anyv1, . . . , vp1,

iv(v1, . . . , vp1) =(v, v1, . . . , vp1) .

In particular, ifp =1, thenivis a scalar

iv =(v)

and ifp =0, then by definition

iv =0 .



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In components,(iv)i1...ip1 =v

j

ji1...ip1.

The interior product is a map

iv : p p1,

or

iv : .

A mapL : is called anderivationif for any p, q,

L( )=

(L) +

L . A mapL : is called ananti-derivationif for any p, q,

L( ) =(L) + (1)p L .

Theorem 3.1.10 LetvE be a vector. The interior product iv : is an anti-derivation.



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3.2 Orientation and the Volume Form

3.2.1 Orientation of a Vector Space

LetEbe a vector space. Let {ei} ={e1, . . . , en} and {ej} ={e1, . . . , en} be twodifferent bases in Erelated by

ei = j

iej,

where =(ij) is a transformation matrix.

Note

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