Boundariesin non–positive curvature

Maciej CzarneckiUniwersytet Lodzki, Katedra Geometrii

Lodz, Polandmaczar@math.uni.lodz.pl

January 10, 2017

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Contents

0 Introduction 3

1 Preliminaries 41.1 Riemannian geometry . . . . . . . . . . . . . . . . . . . . . . . 41.2 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4 Quasi–isometries . . . . . . . . . . . . . . . . . . . . . . . . . 121.5 Model spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.6 Foliations and laminations . . . . . . . . . . . . . . . . . . . . 16

2 Manifolds and spaces of non-positive curvature 192.1 Manifolds of non–positive curvature . . . . . . . . . . . . . . . 192.2 CAT(0) spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3 Gromov hyperbolic spaces . . . . . . . . . . . . . . . . . . . . 24

3 Ideal boundary 263.1 Ideal boundary of Hadamard manifolds . . . . . . . . . . . . . 263.2 Ideal boundary of CAT(0) spaces . . . . . . . . . . . . . . . . 283.3 Continuous extension for quasi–isometries . . . . . . . . . . . 29

4 Contracting boundary 324.1 The Croke–Kleiner example . . . . . . . . . . . . . . . . . . . 324.2 Hyperbolic type geodescics . . . . . . . . . . . . . . . . . . . . 334.3 Contracting boundary for CAT(0) spaces . . . . . . . . . . . . 34

5 Applications to foliations and laminations 355.1 Differential structure . . . . . . . . . . . . . . . . . . . . . . . 355.2 Geometry of leaves . . . . . . . . . . . . . . . . . . . . . . . . 365.3 3–dimensional manifolds . . . . . . . . . . . . . . . . . . . . . 375.4 Remarks on Hadamard laminations . . . . . . . . . . . . . . . 38

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0 Introduction

These notes appeared for support Erasmus+ lectures given by the authorin 2016 in Universidade de Santiago de Compostela and Universidad deGranada.

We generalize the natural boundary of the hyperbolic n–space is the(n − 1)–sphere to non-positively curved manifolds and some metric spaces,namely CAT (0) and δ–hyperbolic. We shall distintc between strictly nega-tive and non–positive case in context of contracting boundary and give someapplications to foliations and laminations.

We start with an introduction to Hadamard manifolds i.e. connected,simply connected and complete Riemannian manifolds of non–positive sec-tional curvature. Then we describe metric spaces CAT (0) (after E. Cartan,A. D. Aleksandrov, V. Toponogov) and Gromov hyperbolic (after E. Ripsand M. Gromov) together with properties similar to manifolds.

Ideal boundary of a non-positively curved space/manifold is representedby ends of geodesic rays. We describe some examples and distinct betweenspaces of non-positive curvature and those of curvature which is negative andbounded from zero. We shall show that quasi–isometric hyperbolic spaceshave homeomorphic ideal boundaries.

Croke and Kleiner showed that the ideal boundary is not a quasi–isometricinvariant in the class of CAT (0) spaces. A new idea of Charney and Sul-tan is a partial solution in this situation. They simply remove geodesics ofnon–hyperbolic type to obtain contracting boundary which has the aboveproperty.

For some regular classes of subspaces like foliations and laminations wegive a few results about boundary behaviour after Fenley, Lee-Yi and theauthor.

I would like to thank Prof. Jesus Alvarez Lopez (USC) and Prof. AntonioMartinez Lopez (UGr) for their help in organizing lectures as well as foroverall hospitality.

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1 Preliminaries

1.1 Riemannian geometry

1.1. Recall that a Riemannian manifold is a differentiable manifold M withtensor g of type (0, 2) (Riemannian metric) which is bilinear, symmetric andpositively definite. In other words, on any tangent space to M we have aninner product.

1.2. We define Levi–Civita connection ∇ on (M, g) by

2 g (∇XY, Z) =Xg(Y, Z) + Y g(Z,X)− Zg(X, Y )

+ g ([X, Y ], Z)− g ([Y, Z], X) + g ([Z,X], Y )

for any vector fields X, Y, Z on M . Here [., .] is the Lie bracket. The Levi–Civita connection is a unique parallel (i.e. ∇g = 0) and torsion–free (i.e.∇XY −∇YX = [X, Y ]) connection on (M, g).

1.3. A length a differentiable curve γ : [a, b] → M is a number l(γ) =∫ ba

√g(c, c). Thus if M is connected (in fact, path connected) then it is

metric space with the disatnce between two points equal to the infimum oflengths of piecewise differentiable curves joining the points.

1.4. The curvature tensor is (1, 3) tensor given by

R(X, Y )Z = ∇X∇YZ −∇Y∇XZ −∇[X,Y ]Z.

If v, w is an orthonormal basis of a 2–dimensional subspace σ of TpM thesectional curvature of M at point p in direction of σ is the number

K(σ) = g(R(v, w)w, v).

1.5. We say that a Riemannian manifold is of constant curvature if thesectional curvature depends neither on point nor on 2-dimensional tangentsubspace at this point.

Respectively, the sectional curvature is bounded from the above by a con-stant κ if it is true at any point and any 2–dimensional direction.

1.6. Examples of constant curvature n–dimensional manifolds are

(i) for K ≡ 0 the Euclidean space En with its standard inner product 〈., .〉,

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(ii) for K ≡ 1 the n–dimensional unit sphere Sn ⊂ Rn+1 with the Rieman-nian metric being restrictions of 〈., .〉,

(iii) for K ≡ −1 the n–dimensional hyperbolic space Hn i.e. the unit ballBn ⊂ Rn with the Riemannian metric given for v, w ∈ TxM by

g(v, w) =4

(1− ‖x‖2)2〈v, w〉.

1.7. On the above manifolds we calculate distance as follows

(i) d(x, y) = ‖x− y‖ in En

(ii) d(x, y) = arccos〈x, y〉 in Sn

(iii)

d(x, y) = 2(tanh)−1

(‖x− y‖√

1− 2〈x, y〉+ ‖x‖2‖y‖2

)in Hn. In case H2 ⊂ C more useful is the formula

d(x, y) = 2(tanh)−1∣∣∣∣ x− y1− xy

∣∣∣∣1.8. A differentiable curve c : I → M on the Riemannian manifold (M, g)is a geodesic if it has no acceleration i.e. ∇c c = 0. Observe that a geodesichas constant (but not necessary unit) speed which means that

√g(c, c) is

constant.

1.9. Every geodesic on a Riemannian manifold locally minimizes distances.Minimazing fails if for instance the geodesic is closed.

1.10. For a unit speed C2 curve γ : (−ε, ε) → M its geodesic curvature atγ(0) is the number

kg(0) =√g (∇γ γ|0,∇γ γ|0).

In H2 any horocycle has geodesic curvature equal 1 while equidistant froma geodesic is of geodesic curvature | cosα| where α is an angle made by theequdistant and bounding circle S1 in the ball model.

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1.11. If ∇ is Levi–Civita connection on a Riemannian manifold (M, g) and Lis a submanifold of M then the connection restricts to Levi-Civita connection∇ on L.

The second fundamental form of L is (1, 2) tensor B given by

B(X, Y ) = ∇X Y −∇XY

for any vector fields X and Y on L; here X and Y are their extensions onM .

1.12. A submanifold is totally geodesic if its second fundamental form van-ishes. In this case, any geodesic on the submanifold is a geodesic on themanifold.

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1.2 Topology

1.13. Let X be a nonempty set and (Yi) a family of topological spaces.Moreover, consider maps fi : Yi → X and gi : X → Yi.

The inverse limit topology in X is the finest topology in which all the fi’sare continuous. We write then X = lim

←−Yi.

Analogously, in X we have the direct limit topology (and then writeX = lim

−→Yi) if this is the coarsest topology in which all the gi’s are con-

tinuous.

1.14. Two continuous cuves σ and τ defined on the interval [0, 1] into atopological space X are homotopic if there is a continuous map H : [0, 1] ×[0, 1]→ X such that H(0, .) = σ, H(1, .) = τ .

1.15. For a path connected topological space X we construct its fundamentalgroup π1(X) taking set of homotopy classes of continuous loops at somex0 ∈ X which we multiply walking along a first loop and then along theother.

We say that a topological space is simply connected if any loop in it istrivial or equivalently π1(X) = 0.

1.16. Let X be a topological space. There is a unique (up to homeomor-

phism) topological space X (called universal cover of X) and a some contin-

uous map f : X → X having the following property

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for any x ∈ X there exists its neighbourhood V , preimage of V is unionof disjoint open sets Uα ⊂ X and f |Uα : Uα → V is a homeomorphism.

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1.3 Metric spaces

Theorem 1.17 (Arzela–Ascoli). Assume that (Y, ρ) is a separable metricspace and (X, d) is compact. The if a sequence fn : Y → X of maps isequicontinuous i.e.

∀ε > 0 ∃δ > 0 ∀n ∈ N ∀y, y′ ∈ Y ρ(y, y′) < δ ⇒ d(fn(y), fn(y′)) < ε,

then some subsequence of (fn) converges uniformly on compact subsets to acontinuous map f : Y → X.

1.18. Let (X, d) and (Y, ρ) be metric spaces. A map f : X → Y is anisometric embedding if it preserves distance i.e.

∀x, x′ ∈ X ρ(f(x), f(x′)) = d(x, x′).

If in additional f is onto then we call it an isometry.

1.19. A metric space X is cocompact if there is its compact subset C ⊂ Msuch that X =

⋃{ϕ(C) | ϕ ∈ Isom (X)}.

1.20. A metric space X is proper if any closed ball in X is compact.

Definition 1.21. A function c : [a, b] → X is a geodesic in X if it is anisometric embedding i.e.

∀t, t′ ∈ [a, b] d(c(t), c(t′)) = |t− t′|.

The same condition we use to define a geodesic ray and a geodesic line (de-fined respectively on half–line [0,∞) or on R).

Definition 1.22. A metric space X is (uniquely) geodesic if any two pointsof X could be joined by a (unique) geodesic.

If geodesic joining p, q ∈ X is unique we simple denote it by [p, q].

1.23. We measure the length of a (continuous) curve γ : [a, b]→ X as

l(γ) = sup

{k−1∑i=0

d(γ(ti), γ(ti+1))

∣∣∣∣∣ a = t0 < t1 < . . . < tk−1 < tk = b

}

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Definition 1.24. For two geodesics c and c′ in X of the same origin p =c(0) = c′(0) define the Aleksandrov angle between them by

^(c, c′) = arccos lim supt,t′→0+

(d(c(t), c′(t′))2 − t2 − t′2

2tt′

The right hand side fraction is the cosine of angle in a Euclidean triangle ofside lengths t = d(p, c(t)), t′ = d(p, c′(t′)) and d(c(t), c′(t′)).

We say that X is a length space if for any two points x, x′ ∈ X there is acurve in X joining them and of length d(x, x′).

1.25. A tubular neighbourhood of radius δ > 0 of a subset A ⊂ X is Nδ(A) ={x ∈ X | d(x,A) < δ}.

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Definition 1.26. The Hausdorff distance between two subsets A,B of ametric space X is

dH(A,B) = inf{δ | A ⊂ Nδ(B) and B ⊂ Nδ(A)}

Definition 1.27. For a finitely generated group G we define its Cayley graphΓG with respect to a finite generating set A taking elements of G as verticesand drawing edge between two vertices iff one is a product of the other byan element from A.

In the Cayley graph we introduce distance — the word metric — makingany edge isometric to the interval [0, 1] and then measuring minimal lengthof cuves along edges from one point to another.

Example 1.28. The Cayley graph of group Z = 〈1〉 is isometric to R andthe word metric is rescticted Euclidean distance.

In the Cayley graph of the free group of two generators F2 = 〈a, b〉 be-tween any two points g, h there is a unique geodesic of length equal to numberof a, a−1, b, b−1 in gh−1 after possible cancellations.

Intuitively, Z has two ”ends” while the set of ends of F2 is the Cantor set{0, 1}N.

Definition 1.29. A uniquely geodesic metric space X is called R–tree if forany x, y, z ∈ X the fact [x, y] ∩ [x, z] = {x} implies [y, z] = [y, x] ∪ [x, z].

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1.4 Quasi–isometries

Definition 1.30. Let (X, d) and (X ′, d′) be metric spaces and λ ≥ 1, ε ≥ 0.A map f : X → X ′ is a (λ, ε)—quasi–isometric embedding if for any x, y ∈ X

1

λd(x, y)− ε ≤ d′(f(x), f(y)) ≤ λd(x, y) + ε.

If, in additional, there is some K such that d′(x′, im f) ≤ K for anyx′ ∈ X ′ then we say that f is a (λ, ε)—quasi–isometry.

A quasi–inverse to a quasi–isometry f : X → X ′ is a (non-unique) mapf ′ : X ′ → X such that there is a constant L

Definition 1.31. A quasi–geodesic (respectively quasi–geodesic ray) is aquasi–isometric embedding of a segment (resp. half–line).

Example 1.32. (i) Every metric space of finite diameter is quasi–isometricto a point.

(ii) A quasi–geodesic (and a quasi–isometry) could very wild. A graph ofa function f : R → [−1, 1] is (1, 1)—quasi–geodesic even if f is notcontinuous.

(iii) Equidistant from a geodesic is a quasi–geodesic with appriopriate con-stants.

Proposition 1.33. For a given finitely generated group its Cayley graphswith respect to any two finite sets of generators, are quasi–isometric.

Proof. Let LC(g) denotes the minimal number of elements of C to expressg ∈ G.

Now let A and B be two finite generating sets for the group G. Takeλ = max{LA(h) | h ∈ B} ·max{LB(K) | k ∈ A}. Then a map sending anyelement of G to itself is (λ, 0)—quasi–isometry.

Proposition 1.34 (taming quasi–geodesics). Let c : [a, b]→ X be a (λ, ε)—quasi–geodesic in a geodesic metric space. Then there is a continuous (λ, ε′)—quasi–geodesic c′ : [a, b]→ X such that

(i) c(a) = c′(a), c(b) = c′(b)

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(ii) ε′ = 2(λ+ ε)

(iii) for any t, t′ ∈ [a, b] the length of c′ on [t, t′] is bounded by k1d(c′(t), c′(t′))+k2 where k1 = λ(λ+ ε), k2 = (λε′ + 3)(λ+ ε)

(iv) the Hausdorff distance dH(im (c), im (c′)) < λ+ ε.

Proof. Let [k, l] = Z ∩ [a, b]. Then the ”broken geodesic” c′ with the image[c(a), c(k)]∪[c(k), c(k+1)]∪. . .∪[c(l), c(b)] satisfies all the conditions. Detailsis [2] Lemma III.H.1.11.

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1.5 Model spaces

1.35. Two–dimensional model geometries of constant curvature κ are

(i) the Euclidean plane E2 when κ = 0

(ii) the 2–dimensional sphere of radius 1√κ

when κ > 0

(iii) the hyperbolic plane rescaled by 1√−κ when κ < 0

We denote them M2κ and call model spaces for appropriate κ’s. The diameter

Dκ of M2κ is infinite for κ ≤ 0 and equals π√

κfor κ > 0.

1.36. For a geodesic triangle in M2κ of side lengths a, b, c (a + b + c < 2Dκ)

and opposite angles α, β, γ the law of cosines is formulated as follows

(i) for κ = 0c2 = a2 + b2 − 2ab cos γ

(ii) for κ > 0

cos(√κc) = cos(

√κa) cos(

√κb) + sin(

√κa) sin(

√κb) cos γ

(iii) for κ < 0

cosh(√−κc) = cosh(

√−κa) cosh(

√−κb)−sinh(

√−κa) sinh(

√−κb) cos γ

Anytime, c is an increasing function of γ.

1.37. Images of geodesics on M2κ are Euclidean segments in E2, arcs of great

circles for κ > 0 and arcs of circles ortogonal to the boundary circle for κ < 0.

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Theorem 1.38 (Aleksandrov Lemma). Consider a geodesic triangle ∆ABCin M2

κ of sides a, b, c and angles α, β, γ.Let B′ be such a point that B and B′ lie on opposite side of the line

through A and C and γ + γ′ ≥ π where α′, β′, γ′ are respective angles anda′, b, c′ sides of the geodesic triangle ∆AB′C .

Then a + a′ ≤ c + c′ and angles of a triangle with sides a + a′, c, c′ arerespectively greater or equal to angles α + α′, β, β′.

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1.6 Foliations and laminations

Definition 1.39. Let M be a differentiable manifold of dimension n. A p–dimensional foliation (or more often a foliation of codimension q = n− p) ofclass Cr is a family of foliated charts (ϕi) i.e. maps ϕi : Ui → Bi×Di whereUi is open in M and Bi, Di are balls in Rp and Rq respectively such that

(F1)⋃Ui = M

(F2) ∀ i, j the map ϕj ◦ ϕ−1i is of class Cr and its last q coordinates do notdepend on the last q coordinates of the argument.

A maximal union of non–disjoint subsets of the form Bi × {point} is calleda leaf of the foliation.

Definition 1.40. Let X be a topological space. A p–dimensional laminationin X is a family maps ϕi : Ui → Bi × Ti where Ui is open in M and Bi is aball in Rp while Ti is some topological space such that

(L1) ∀ i, j the map ϕj ◦ ϕ−1i and its coordinate coming from Tj does notdepend of the coordinate of the argument coming from Ti.

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(L2) union of all leaves (defined as for foliations) is a closed set in X

Example 1.41. The classical Reeb foliation is a foliation of 3–sphere whoseone leaf is a torus (Clifford torus) and both remaining domain (open solidtori) are filled with topological planes.

The Reeb component inside the torus is constructed as follows. Familyof graph of functions fb : (−1, 1) 3 x 7→ ex(x2 − 1) + b = y with b ∈ R isrotated around y–axis and then quotient by vertical action of Z (translations)is taken.

Theorem 1.42 (Novikov). Any C2 codimension 1 foliation of the sphere S3

contains a compact leaf. This leaf is a topological torus and in its interiorthe foliation is Reeb component.

There are many geometric obstruction for existence particular foliations.For instance, on compact hyperbolic manifolds there no geometric foliation in

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any reasonable sense (totally geodesic, totally umbilical, Riemannian, quasi–isometric etc.). Some details and references could be found in [7].

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2 Manifolds and spaces of non-positive cur-

vature

2.1 Manifolds of non–positive curvature

Definition 2.1. Hadamard manifold is a connected, simply connected Rie-mannian complete manifold of non-positive curvature.

Theorem 2.2 (Hadamard–Cartan). If M is an Hadamard manifold then forany p ∈M the exponential map expp : TpM →M is a diffeomorphism.

Thus n–dimensional Hadamard manifold is diffeomorphic to open n–ball.

Corollary 2.3. For any two points p, q in an Hadamard manifold M thereis unique unit–speed geodesic joining p to q.

Example 2.4. On torus T 2 = S1×S1 there is a flat (i.e. K ≡ 0) Riemannianmetric induced from the universal cover R2.

Every compact genus g ≥ 2 surface Σg carries hyperbolic (i.e. K ≡ −1)Riemannian metric induced from the universal cover H2.

2.5. In a geodesic triangle of sides a, b, c and opposite angles α, β, γ on anHadamard manifold trigonometric inequalities hold

law of cosines c2 ≥ a2 + b2 − 2ab cos γ

double law of cosines c ≤ a cos β + b cosα

angle sum α + β + γ ≤ π

2.6. A function f a Riemannian manifold (M, g) is convex if for any maximalgeodesic γ on M the function f ◦ γ : R→ R is convex in the usual sense. InC2 case this means that 2–form given by (∇2f)p (v, w) = g (∇vgrad f, w) ispositively semi–definite at any p ∈M .

On Hadamard manifolds the following functions are convex:

distance from a closed convex subset,

distance from a complete totally geodesic submanifold.

2.7. For p, q, r ∈ M we denoted by ^p(q, r) the angle at p subtended by qand r i.e. angle between geodesic starting at p in directions to q and r.

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Definition 2.8. We say that an Hadamard manifold M satisfies the visibilitycondition (or simply is visible) if for any point p ∈M and any ε > 0 there issuch R that any maximal geodesic γ of distance ≥ R from p is visible underangle ≤ ε i.e.

^p(γ) = sup{^p(γ(t), γ(s)) | t, s ∈ R} ≤ ε.

More informally, M is visible if distant geodesic lines look small.

M is uniformly visible if in addition, R does not depend on p.

Theorem 2.9. If an Hadamard manifold is of curvature bounded from theabove by κ < 0 then it is uniformly visible.

Theorem 2.10. Let M be a cocompact Hadamard manifold. Then M isvisible iff M admits no totally geodesic 2–submanifold isometric (in inducedmetric) to the Euclidean plane E2.

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2.2 CAT(0) spaces

Definition 2.11. A geodesic metric space (X, d) is a CAT(κ) space if forany geodesic triangle ∆pqr in X its comparison triangle ∆pqr in M2

κ (i.e.geodesic triangle of the same side lengths) has the following property

(CAT) For any x ∈ [p, q], y ∈ [p, r] their comparison points x ∈ [p, q], y ∈ [p, r]being at the same distance from p as x and y from p, satisfy

d(x, y) ≤ d(x, y).

Definition 2.12. We call a geodesic metric space an Hadamard space it isCAT (0) and complete.

Proposition 2.13. For a geodesic metric space X the following are equiva-lent

(i) X is CAT(κ) space.

(ii) For any geodesic triangle ∆ a median (i.e. geodesic segment joining avertex with midpoint of opposite side) in comparison triangle ∆ is notshorter than corresponding median in ∆.

(iii) For any geodesic triangle ∆ Aleksandrov angles in its comparison tri-angle ∆ are not less than corresponding angles in ∆.

Remark 2.14. Condition (iii) from 2.13 allows to to express CAT(0) def-inition in a purely distance way. CN inequality of Bruhat–Tits says that

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a geodesic metric space (X, d) is CAT (0) iff for any p, q, r ∈ X and anymidpoint m of q and r yields

(d(p, q))2 + (d(p, r))2 ≥ 2(d(p,m))2 +1

2(d(q, r))2.

2.15. Observe (not trivial cf.[2]) that CAT(κ) implies CAT(κ′) for κ < κ′.Moreover, CAT(0) spaces are very simple from topological point of view.

They are contractible so in particular simply connected.

Example 2.16. (i) Simply connected Riemannian manifold of curvaturebounded by κ from the above is a CAT(κ) space.

(ii) Convex subset of a CAT(κ) space is CAT(κ) itself.

(iii) Rn and Hn are CAT(0) but only Hn is CAT(−1).

(iv) R–tree (cf. 1.29) is a CAT(κ) space for any κ.

(v) E2 with open quadrant removed (and length metric) is a CAT(0) space.Geodesic in such a space are Euclidean segments (if possible) or unionsof two Euclidean segments with one end 0. Hence it is enough to useAleksandrov Lemma 1.38 and 2.13 (iii).

(vi) E3 with open ”octave” removed (and length metric) is not a CAT(0)space because its contains a geodesic triangle with three right angles.

Theorem 2.17 (Hadamard–Cartan for CAT(κ)). If a geodesic metric spaceX has curvature bounded by κ from the above with κ ≤ 0 then its universalcover X is a CAT (κ) space.

2.18. The distance in CAT(0) space is convex in the following sense. Ifc : [0, 1] → X and c′ : [0, 1] → X are geodesics parametrized proportionallyto arc–length in a CAT(0) space then for any t ∈ [0, 1]

d(c(t), c′(t)) ≤ (1− t) d(c(0), c′(0)) + t d(c(1), c′(1)).

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Definition 2.19. Let C be a convex and complete subset in a CAT(0) space(X, d). For x ∈ X its projection onto C is the unique point πC(x) realizingdistance d(x,C).

Points of the geodesic segment [x, πC(x)] project onto πC(x).

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2.3 Gromov hyperbolic spaces

Definition 2.20. We say that a geodesic triangle in a metric space X con-sisting of geodesics segments [x, y], [x, z], [y, z] is δ–slim if every its side isin δ–neighbourhood of other two sides i.e. [x, y] ⊂ Nδ ([x, z] ∪ [y, z]) etc. A

geodesic metric space X is δ–hyperbolic if every geodesic triangle in X isδ–slim

Definition 2.21. We define the Gromov product of points x, y ∈ X withrespect to w ∈ X as

(x, y)w =1

2(d(x,w) + d(y, w)− d(x, y)).

A metric space is δ–hyperbolic iff for any x, y, z, w

(x, y)w ≥ min((x, z)w, (y, z)w)− δ.

Theorem 2.22. If space X is CAT(κ) for some κ ≤ 0 then X is δ–hyperbolicfor some δ.

Example 2.23. (i) H2 is (ln(1 +√

2))–hyperbolic.

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(ii) E2 is not hyperbolic because in big triangles their sides are far fromeach other.

(iii) R–tree is 0–hyperbolic.

Theorem 2.24. A proper CAT(0) space is δ–hyperbolic for some δ iff it isuniformly visible (cf. 2.8).

Theorem 2.25 (Flat Plane Theorem). A proper cocompact CAT(0) space isδ–hyperbolic for some δ iff it does not contain a metric subspace isometric toE2 (in induced metric).

Definition 2.26. A group G is a negatively curved group if it is finitelygenerated and its Cayley graph is δ–hyperbolic for some δ.

Such a group is non-elementary negatively curved group if it is infiniteand is not a finite extension of cyclic group.

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3 Ideal boundary

3.1 Ideal boundary of Hadamard manifolds

Definition 3.1. Let M be an Hadamard manifold. Its ideal boundary M(∞)consists of classes of geodesic rays in M with respect to the relation of beingasymptotic γ ∼ τ iff dH(im (γ), im (τ)) <∞.

3.2. For any p ∈ M and any geodesic ray γ in M there is unique geodesicray γp,z : [0,∞)→M such that γp,z(0) = p and γp,z ∼ γ.

Definition 3.3. In M = M ∪M(∞) we define cone topology whose basisare truncated cones

T (v, ε, r) ={z ∈ M | ^(γv, γp,z) < ε

}\ B(p, r)

where v ∈ T 1pM and γv denotes geodesic ray starting at p in direction of v.

3.4. The cone topology on M is admissible i.e.

M is dense in M and topology induced on M is its original one,

for any geodesic ray γ : [0,∞) → M its extension γ : [0,∞] → M byγ(∞) = [γ] is continuous,

for any ϕ ∈ Isom (M) its extension ϕ : M → M by ϕ([γ]) = [ϕ ◦ γ] forany geodesic ray, is a homeomorphism.

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3.5. For any p ∈ M there natural one-to-one correspondence between unittangent spce T 1

pM and M(∞) (a vector is mapped onto geodesic ray in thisdirection). This correspondence is a homeomorphism so M(∞) is homeo-morphic to Sn−1 if dimM = n. Only in some special cases like e.g. constantcurvature we have a differential structure on M(∞) being extension of thaton M .

Proposition 3.6. If M satisfies the visibility condition then any two distinctpoints of M(∞) could be joined by unique geodesic line in M .

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3.2 Ideal boundary of CAT(0) spaces

Since the notion of geodesic ray and the relation of being asymptotic areformulated in a purely metric way we can define the ideal boundary of CAT(0)space as in 3.1.

Theorem 3.7. If X is a proper CAT (0) space then

(i) the inclusion X ⊂ X is a homeomorphism onto image

(ii) ∂X is compact

(iii) X is compact

Tits metric

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3.3 Continuous extension for quasi–isometries

Proposition 3.8 (logarithmic stability of curves). Let (X, d) be a δ-hyperbolicgeodesic space and c a rectifiable path in X with ends p and q. Then for anyx ∈ [p, q]

d(x, im (c)) ≤ δ | log2 l(c)|+ 1

Proof. Let c : [a, b] → X be a path of finite length. For x ∈ [p, q] we findclosest points not on image of path c but on geodesic segments joining pointson c. Using δ–slimness we find a point y1 ∈

[c(a), c

(a+b2

)]∪[c(a+b2

), c(b)

]such that d(x, y1) ≤ δ and then process dividing intervals into halves up totheir ends are of distance close to 1. Details in [2] Proposition III.H.1.6.

Theorem 3.9 (stability of quasi–geodesics). Assume that δ ≥ 0, λ ≥ 1 andε ≥ 0. There is such R > 0 that if (X, d) is a δ-hyperbolic geodesic space andc a (λ, ε)—quasi–geodesic with ends p and q then

dH ([p, q], im (c)) ≤ R.

Proof. We tame c as c′ as in 1.34. Then dH(im (c), im (c′)) ≤ λ + ε. Thenfor D being maximal distance from points of [p, q] to c′ and x0 realizing thismaximum we construct a curve which allows by 3.8 and 1.34 to find D0 whichdepends only on λ, ε and δ.

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The geodesic segment [p, q] is then in D0 neighbourhood of im (c′) andonly uniformly short part of im (c′) is outside ND0([p, q]). Taming once againgives demanded R. Details in [2] Theorem III.H.1.7.

Corollary 3.10. A geodesic metric space X ′ which could be quasi–isometricallyembedded in a δ–hyperbolic geodesic space X, is δ′–hyperbolic for some δ′.

Proof. It is enough to check slimness of quasi–geodesic triangles.

Proposition 3.11 (geodesic companion). If c is a quasi–geodesic ray in aproper δ–hyperbolic geodesic space (X, d) then there is a geodesic ray γ in Xwhich asymptotic to c i.e. dH (im (c), im (γ)) is finite.

Proof. Let γn be the geodesic ray through c(0) and c(n). X is proper thus Xis compact and γn are 1-equicontinuous so by Arzela–Ascoli theorem there a

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limit (even for all this sequence) which is in fact a geodesic ray asymptoticto c.

For a quasi–geodesic ray c : [0,∞]→ X we write rather c(∞) than [c] todenote the asymptoticity class of geodesics in X which are asymptotic to c.

Theorem 3.12 (continuous extension for quasi–isometries). Let X1 and X2

be proper δ–hyperbolic geodesic spaces. If f : X1 → X2 is a quasi–isometricembedding then ∂f : ∂X1 → ∂X2 defined as ∂f([γ]) = (f ◦ γ)(∞), is atopological embedding.

In particular, quasi–isometric hyperbolic spaces have homeomorphic idealboundaries.

Proof. [2] III.H.3.9

Example 3.13. The above properties are not true in case of CAT (0) spaces.Semicircle in E2 is a quasi–geodesic but its Haudorff distance from the

diameter is linear function of the length. Some spiral in E2 is a quasi–geodesicbut has no end at infinity.

Losing continuity of quasi–isometries on the ideal boundary is more so-phisticated but also true (cf. [5] and the next section).

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4 Contracting boundary

4.1 The Croke–Kleiner example

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4.2 Hyperbolic type geodescics

Let (X, d) be a geodesic metric space.

Definition 4.1. A geodesic γ in X is D–contracting if for any x, y ∈ X suchthat d(x, y) < d(x, πγ(x)) the distance d(πγ(x), πγ(y)) ≤ D.

Definition 4.2. A geodesic γ in X is Morse if for any K ≥ 1 and L ≥ 0there is such M that any (K,L)—quasi–geodesic with ends on im (γ) lies inNM(γ). In this case we say γ is M–Morse where M is a nonnegative functionon [1,∞)× [0,∞).

Definition 4.3. A geodesic γ in X is δ–slim if for any x ∈ X and any y, z ∈im (γ) every point w of the geodesic [y, z] satisfies d(w, [x, y] ∪ [x, z]) ≤ δ.

Theorem 4.4 (Charney–Sultan [4]). For a geodesic ray γ in a CAT (0) spaceX the following are equivalent

(i) γ is contracting

(ii) γ is Morse

(iii) γ is slim

Example 4.5. In H2 any geodesic line is contracting. The maximum oflength of projection segment realizes a horocycle tangent to the line. ThusD = 2 ln(

√2 + 1).

On the other hand, in E2 there are no contracting lines because balls ofradius r project onto segments of length 2r.

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4.3 Contracting boundary for CAT(0) spaces

Croke and Kleiner showed the ideal boundary is not a quasi–isometric in-variant in the class of CAT (0) spaces. A new idea of Charney and Sul-tan is a partial solution in this situation. They simply remove geodesics ofnon–hyperbolic type to obtain contracting boundary which has the aboveproperty.

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5 Applications to foliations and laminations

5.1 Differential structure

The ideal boundary of an Hadamard manifold is homeomorphic to spherebut in general, there is no differential structure on the boundary extendingdifferential structure of the manifold. This occurs in some particular caseslike for instance constant curvature.

Totally geodesic foliations of Hn were classified by Ferus as those havingorthogonal transversal of geodesic curvature ≤ 1.

Theorem 5.1 (Lee–Yi [12]). All the totally geodesic Ck foliations of Hn butnot orthogonal to horocycles are in one-to-one correspondce (modulo isomet-ric action of the group O(n−1)×R×Z2) with Ck−1 functions z : [0, π]→ Sn−1such that z(0) = z(π) = (1, 0, . . . 0) and ‖z′‖ ≤ 1.

For a totally geodesic foliation F of Hn function z is built as follows. Let0 /∈ L ∈ F . Then z(r) is the spherical center of the subsphere L(∞) inSn−1 = Hn(∞) and r is spherical radius of L(∞).

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5.2 Geometry of leaves

The norm of the second fundamental form of a submanifold L in (M, g) isthe least upper bound ‖BL‖ of maxima of the quadratic form B on any unittangent space to L.

Definition 5.2. If F is a foliation of a Riemanian manifold M the norrm ofthe second fundamental form of foliation F we call

‖BF‖ = sup{‖BL‖ | L ∈ F}.

Theorem 5.3 (Czarnecki [6]). Assume that F is a C2 foliation of Hn with‖BF‖ < 1. Then

(i) all the leaves of F are Hadamard manifolds

(ii) there is a canonical continuous embedding of the union of leaf idealboundaries into Hn(∞) given by [γ] 7→ γ(∞) for any geodesic ray γon leaf. Here the topology on the union of leaf ideal boundaries comesfrom the projection of the unit tangent bundle to F onto

⋃L(∞).

In codimension 1, a condition for separating leaf boundaries is an estimateof normal curvature of F by ‖BF‖.

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5.3 3–dimensional manifolds

5.4. If 3–manifold has non–elementary negatively curved fundamental group(cf. 2.26) then it admiits hyperbolic Riemannian structure. In particular,such a manifold has universal cover which is H3 and we could study its idealboundary.

Theorem 5.5 (Fenley [10]). Let M be a closed irreducible 3–manifold withnon–elementary negatively curved fundamental group, F a codimension 1Reebless foliations and F the lift of F to the universal cover.

Then either limit set of any leaf its limit set is S2 = H3(∞) or of any leafits limit set is is not equal to S2.

More sophisticated is newer result. It is closely related to extension onideal boundary.

Theorem 5.6 (Fenley [11]). If a foliation F on a 3–dimensional atoroidalclosed manifold M is almost transverse to a quasi-geodesic pseudo–Anosovflow then π1(M) is negatively curved and F has continuous extension prop-erty i.e. ideal boundary of leaf universal cover embed topologically into idealboundary of M .

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5.4 Remarks on Hadamard laminations

All the previous consideration

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References

[1] W. Ballmann, Lectures on Spaces of Nonpositive Curvature, Birhauser1995.

[2] M. Bridson, A. Haefliger, Metric Spaces of Non-Positive Curvature,Springer 1999.

[3] A. Candel, L. Conlon, Foliations I and Foliations II, American Mathe-matical Society 2001 and 2003.

[4] R. Charney, H. Sultan, Contracting boundaries of CAT(0) spaces, J.Topol. 8 (2015), 93–117.

[5] C. B. Croke, B. Kleiner, Spaces with nonpositive curvature nad theirideal boundaries, Topology 39 (2000) 549–556.

[6] M. Czarnecki, Hadamard foliations of Hn, Diff. Geom. Appl. 20 (2004),357–365.

[7] M. Czarnecki, P. Walczak, Extrinsic geometry of foliations in Foliations2005, 149-167, World Scientific Publishers 2006.

[8] P. Eberlein, B. O’Neill, Visibility manifolds, Pacific J. Math. 46 (1973),45–109.

[9] P. Eberlein, Geometry of Nonpositively Curved Manifolds, University ofChicago Press 1996.

[10] S. Fenley, Limit sets of foliations in hyperbolic 3–manifolds, Topology37 (1998), 875–894.

[11] S. Fenley, Geometry of foliations and flows. I. Almost transverse pseudo–Anosov flows and asymptotic behavior of foliations, J. Differential Geom.81 (2009), 1–89.

[12] K. B. Lee, S. Yi, Metric foliations on hyperbolic spaces, J. Korean Math.Soc. 48 (1) (2011), 63–82.

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