in non–positive curvature

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Boundaries

in non–positive curvature

Maciej Czarnecki

Uniwersytet L´ odzki, Katedra Geometrii L´ od´ z, Poland

maczar@math.uni.lodz.pl

January 10, 2017

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0 Introduction

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

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 negative and non–positive case in context of contracting boundary and give some applications to foliations and laminations.

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

Ideal boundary of a non-positively curved space/manifold is represented by ends of geodesic rays. We describe some examples and distinct between spaces of non-positive curvature and those of curvature which is negative and bounded from zero. We shall show that quasi–isometric hyperbolic spaces have homeomorphic ideal boundaries.

Croke and Kleiner showed that the ideal boundary is not a quasi–isometric invariant 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 of non–hyperbolic type to obtain contracting boundary which has the above property.

For some regular classes of subspaces like foliations and laminations we give a few results about boundary behaviour after Fenley, Lee-Yi and the author.

I would like to thank Prof. Jesús Álvarez López (USC) and Prof. Antonio Martinez López (UGr) for their help in organizing lectures as well as for overall hospitality.

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1 Preliminaries

1.1 Riemannian geometry

1.1. Recall that a Riemannian manifold is a differentiable manifold M with tensor g of type (0, 2) (Riemannian metric) which is bilinear, symmetric and positively definite. In other words, on any tangent space to M we have an inner 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) = Rb

apg( ˙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 of lengths 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 T_pM the sectional 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 the sectional curvature depends neither on point nor on 2-dimensional tangent subspace at this point.

Respectively, the sectional curvature is bounded from the above by a constant 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 Eⁿwith its standard inner product h., .i,

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

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

g(v, w) = 4

(1 − kxk²)²hv, wi.

1.7. On the above manifolds we calculate distance as follows (i) d(x, y) = kx − yk in Eⁿ

(ii) d(x, y) = arccoshx, yi in Sⁿ (iii)

d(x, y) = 2(tanh)⁻¹ kx − yk

p1 − 2hx, yi + kxk²kyk²

!

in Hⁿ. In case H² ⊂ C more useful is the formula d(x, y) = 2(tanh)⁻¹

x − y 1 − x¯y

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 geodesic has constant (but not necessary unit) speed which means that pg( ˙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 at γ(0) is the number

k_g(0) = q

g (∇_γ_˙˙γ|₀, ∇_γ_˙˙γ|₀).

In H² any horocycle has geodesic curvature equal 1 while equidistant from a geodesic is of geodesic curvature | cos α| where α is an angle made by the equdistant 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 L is 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 on M .

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 the manifold.

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1.2 Topology

1.13. Let X be a nonempty set and (Y_i) a family of topological spaces.

Moreover, consider maps f_i : Y_i → X and g_i : X → Y_i.

The inverse limit topology in X is the finest topology in which all the f_i’s are continuous. We write then X = lim

←−Y_i.

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

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

1.14. Two continuous cuves σ and τ defined on the interval [0, 1] into a topological 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 fundamental group π₁(X) taking set of homotopy classes of continuous loops at some x₀ ∈ X which we multiply walking along a first loop and then along the other.

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

1.16. Let X be a topological space. There is a unique (up to homeomorphism) topological space eX (called universal cover of X) and a some continuous map f : eX → X having the following property

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

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1.3 Metric spaces

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

∀ε > 0 ∃δ > 0 ∀n ∈ N ∀y, y⁰ ∈ Y ρ(y, y⁰) < δ ⇒ d(f_n(y), f_n(y⁰)) < ε, then some subsequence of (f_n) converges uniformly on compact subsets to a continuous map f : Y → X.

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

∀x, x⁰ ∈ X ρ(f (x), f (x⁰)) = d(x, x⁰).

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 ⊂ M such that X = S{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 an isometric embedding i.e.

∀t, t⁰ ∈ [a, b] d(c(t), c(t⁰)) = |t − t⁰|.

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

Definition 1.22. A metric space X is (uniquely) geodesic if any two points of 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

X

i=0

d(γ(t_i), γ(t_i+1))

a = t₀ < t₁ < . . . < t_k−1 < t_k = b )

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Definition 1.24. For two geodesics c and c⁰ in X of the same origin p = c(0) = c⁰(0) define the Aleksandrov angle between them by

^(c, c⁰) = arccos lim sup

t,t⁰→0⁺

(d(c(t), c⁰(t⁰))²− t² − t⁰² 2tt⁰

The right hand side fraction is the cosine of angle in a Euclidean triangle of side lengths t = d(p, c(t)), t⁰ = d(p, c⁰(t⁰)) and d(c(t), c⁰(t⁰)).

We say that X is a length space if for any two points x, x⁰ ∈ X there is a curve in X joining them and of length d(x, x⁰).

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

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

d_H(A, B) = inf{δ | A ⊂ N_δ(B) and B ⊂ N_δ(A)}

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

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

Example 1.28. The Cayley graph of group Z = h1i is isometric to R and the word metric is rescticted Euclidean distance.

In the Cayley graph of the free group of two generators F₂ = ha, bi between any two points g, h there is a unique geodesic of length equal to number of 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 for any 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 (X⁰, d⁰) be metric spaces and λ ≥ 1, ε ≥ 0.

A map f : X → X⁰ is a (λ, ε)—quasi–isometric embedding if for any x, y ∈ X 1

λd(x, y) − ε ≤ d⁰(f (x), f (y)) ≤ λd(x, y) + ε.

If, in additional, there is some K such that d⁰(x⁰, im f ) ≤ K for any x⁰ ∈ X⁰ then we say that f is a (λ, ε)—quasi–isometry.

A quasi–inverse to a quasi–isometry f : X → X⁰ is a (non-unique) map f⁰ : X⁰ → X such that there is a constant L

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

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

(ii) A quasi–geodesic (and a quasi–isometry) could very wild. A graph of a function f : R → [−1, 1] is (1, 1)—quasi–geodesic even if f is not continuous.

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

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

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

Now let A and B be two finite generating sets for the group G. Take λ = max{LA(h) | h ∈ B} · max{LB(K) | k ∈ A}. Then a map sending any element 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, ε⁰)—

quasi–geodesic c⁰ : [a, b] → X such that (i) c(a) = c⁰(a), c(b) = c⁰(b)

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(ii) ε⁰ = 2(λ + ε)

(iii) for any t, t⁰ ∈ [a, b] the length of c⁰ on [t, t⁰] is bounded by k₁d(c⁰(t), c⁰(t⁰))+

k₂ where k₁ = λ(λ + ε), k₂ = (λε⁰+ 3)(λ + ε) (iv) the Hausdorff distance d_H(im (c), im (c⁰)) < λ + ε.

Proof. Let [k, l] = Z ∩ [a, b]. Then the ”broken geodesic” c⁰ with the image [c(a), c(k)]∪[c(k), c(k +1)]∪. . .∪[c(l), c(b)] satisfies all the conditions. Details is [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 M_κ² and call model spaces for appropriate κ’s. The diameter D_κ of M_κ² is infinite for κ ≤ 0 and equals ^√^π_κ for κ > 0.

1.36. For a geodesic triangle in M_κ² of side lengths a, b, c (a + b + c < 2D_κ) and opposite angles α, β, γ the law of cosines is formulated as follows

(i) for κ = 0

c² = 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 M_κ² 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 ∆ABC in M_κ² of sides a, b, c and angles α, β, γ.

Let B⁰ be such a point that B and B⁰ lie on opposite side of the line through A and C and γ + γ⁰ ≥ π where α⁰, β⁰, γ⁰ are respective angles and a⁰, b, c⁰ sides of the geodesic triangle ∆AB⁰C .

Then a + a⁰ ≤ c + c⁰ and angles of a triangle with sides a + a⁰, c, c⁰ are respectively greater or equal to angles α + α⁰, β, β⁰.

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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) of class C^r is a family of foliated charts (ϕ_i) i.e. maps ϕ_i : U_i → B_i× D_i where U_i is open in M and B_i, D_i are balls in R^p and R^q respectively such that

(F1) S U_i = M

(F2) ∀ i, j the map ϕ_j ◦ ϕ⁻¹_i is of class C^r and its last q coordinates do not depend on the last q coordinates of the argument.

A maximal union of non–disjoint subsets of the form B_i × {point} is called a leaf of the foliation.

Definition 1.40. Let X be a topological space. A p–dimensional lamination in X is a family maps ϕ_i : U_i → B_i× T_i where U_i is open in M and B_i is a ball in R^p while T_i is some topological space such that

(L1) ∀ i, j the map ϕ_j ◦ ϕ⁻¹_i and its coordinate coming from T_j does not depend of the coordinate of the argument coming from T_i.

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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 whose one leaf is a torus (Clifford torus) and both remaining domain (open solid tori) are filled with topological planes.

The Reeb component inside the torus is constructed as follows. Family of graph of functions f_b : (−1, 1) 3 x 7→ e^x(x² − 1) + b = y with b ∈ R is rotated 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 interior the foliation is Reeb component.

There are many geometric obstruction for existence particular foliations.

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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 for any p ∈ M the exponential map exp_p : T_pM → 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 there is unique unit–speed geodesic joining p to q.

Example 2.4. On torus T² = S¹×S¹there is a flat (i.e. K ≡ 0) Riemannian metric induced from the universal cover R².

Every compact genus g ≥ 2 surface Σ_g 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 an Hadamard manifold trigonometric inequalities hold

law of cosines c² ≥ a²+ b²− 2ab cos γ double law of cosines c ≤ a cos β + b cos α angle sum α + β + γ ≤ π

2.6. A function f a Riemannian manifold (M, g) is convex if for any maximal geodesic γ on M the function f ◦ γ : R → R is convex in the usual sense. In C² case this means that 2–form given by (∇²f )_p(v, w) = g (∇vgrad f, w) is positively 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 q

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Definition 2.8. We say that an Hadamard manifold M satisfies the visibility condition (or simply is visible) if for any point p ∈ M and any ε > 0 there is such R that any maximal geodesic γ of distance ≥ R from p is visible under angle ≤ 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 the above by κ < 0 then it is uniformly visible.

Theorem 2.10. Let M be a cocompact Hadamard manifold. Then M is visible iff M admits no totally geodesic 2–submanifold isometric (in induced metric) 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 for any geodesic triangle ∆pqr in X its comparison triangle ∆¯p¯q¯r in M_κ² (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 is CAT (0) and complete.

Proposition 2.13. For a geodesic metric space X the following are equivalent

(i) X is CAT(κ) space.

(ii) For any geodesic triangle ∆ a median (i.e. geodesic segment joining a vertex with midpoint of opposite side) in comparison triangle ¯∆ is not shorter than corresponding median in ∆.

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

Remark 2.14. Condition (iii) from 2.13 allows to to express CAT(0) def-

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a geodesic metric space (X, d) is CAT (0) iff for any p, q, r ∈ X and any midpoint 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(κ⁰) for κ < κ⁰. 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 curvature bounded 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 unions of two Euclidean segments with one end 0. Hence it is enough to use Aleksandrov 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 space X has curvature bounded by κ from the above with κ ≤ 0 then its universal cover ˜X is a CAT (κ) space.

2.18. The distance in CAT(0) space is convex in the following sense. If c : [0, 1] → X and c⁰ : [0, 1] → X are geodesics parametrized proportionally to arc–length in a CAT(0) space then for any t ∈ [0, 1]

d(c(t), c⁰(t)) ≤ (1 − t) d(c(0), c⁰(0)) + t d(c(1), c⁰(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 π_C(x) realizing distance d(x, C).

Points of the geodesic segment [x, π_C(x)] project onto π_C(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 is in δ–neighbourhood of other two sides i.e. [x, y] ⊂ N_δ([x, z] ∪ [y, z]) etc. A

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

Definition 2.21. We define the Gromov product of points x, y ∈ X with respect 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–hyperbolic for 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 from each other.

(iii) R–tree is 0–hyperbolic.

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

Theorem 2.25 (Flat Plane Theorem). A proper cocompact CAT(0) space is δ–hyperbolic for some δ iff it does not contain a metric subspace isometric to E² (in induced metric).

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

Such a group is non-elementary negatively curved group if it is infinite and 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 being asymptotic γ ∼ τ iff d_H(im (γ), im (τ )) < ∞.

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

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

T (v, ε, r) =z ∈ ¯M | ^(γv, γ_p,z) < ε \ ¯B(p, r)

where v ∈ T_p¹M and γ_v 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 by γ(∞) = [γ] is continuous,

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

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3.5. For any p ∈ M there natural one-to-one correspondence between unit tangent spce T_p¹M and M (∞) (a vector is mapped onto geodesic ray in this direction). This correspondence is a homeomorphism so M (∞) is homeomorphic to Sⁿ⁻¹ if dim M = n. Only in some special cases like e.g. constant curvature we have a differential structure on M (∞) being extension of that on M .

Proposition 3.6. If M satisfies the visibility condition then any two distinct points 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 are formulated 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-hyperbolic geodesic space and c a rectifiable path in X with ends p and q. Then for any x ∈ [p, q]

d(x, im (c)) ≤ δ | log₂l(c)| + 1

Proof. Let c : [a, b] → X be a path of finite length. For x ∈ [p, q] we find closest points not on image of path c but on geodesic segments joining points on 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 to their 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 and ε ≥ 0. There is such R > 0 that if (X, d) is a δ-hyperbolic geodesic space and c a (λ, ε)—quasi–geodesic with ends p and q then

dH([p, q], im (c)) ≤ R.

Proof. We tame c as c⁰ as in 1.34. Then d_H(im (c), im (c⁰)) ≤ λ + ε. Then for D being maximal distance from points of [p, q] to c⁰ and x₀ realizing this maximum we construct a curve which allows by 3.8 and 1.34 to find D0 which depends only on λ, ε and δ.

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The geodesic segment [p, q] is then in D₀ neighbourhood of im (c⁰) and only uniformly short part of im (c⁰) is outside N_D₀([p, q]). Taming once again gives demanded R. Details in [2] Theorem III.H.1.7.

Corollary 3.10. A geodesic metric space X⁰ which could be quasi–isometrically embedded in a δ–hyperbolic geodesic space X, is δ⁰–hyperbolic for some δ⁰. Proof. It is enough to check slimness of quasi–geodesic triangles.

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

Proof. Let γ_nbe the geodesic ray through c(0) and c(n). X is proper thus ¯X is compact and γ_n 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 asymptotic to c.

For a quasi–geodesic ray c : [0, ∞] → X we write rather c(∞) than [c] to denote 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–isometric embedding then ∂f : ∂X1 → ∂X2 defined as ∂f ([γ]) = (f ◦ γ)(∞), is a topological embedding.

In particular, quasi–isometric hyperbolic spaces have homeomorphic ideal boundaries.

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–geodesic but has no end at infinity.

Losing continuity of quasi–isometries on the ideal boundary is more sophisticated 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 such that d(x, y) < d(x, π_γ(x)) the distance d(π_γ(x), π_γ(y)) ≤ D.

Definition 4.2. A geodesic γ in X is Morse if for any K ≥ 1 and L ≥ 0 there is such M that any (K, L)—quasi–geodesic with ends on im (γ) lies in N_M(γ). In this case we say γ is M –Morse where M is a nonnegative function on [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) space X 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 of length of projection segment realizes a horocycle tangent to the line. Thus D = 2 ln(√

2 + 1).

On the other hand, in E² there are no contracting lines because balls of radius 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 invariant 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 of non–hyperbolic type to obtain contracting boundary which has the above property.

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5 Applications to foliations and laminations

5.1 Differential structure

The ideal boundary of an Hadamard manifold is homeomorphic to sphere but in general, there is no differential structure on the boundary extending differential structure of the manifold. This occurs in some particular cases like for instance constant curvature.

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

Theorem 5.1 (Lee–Yi [12]). All the totally geodesic C^k foliations of Hⁿ but not orthogonal to horocycles are in one-to-one correspondce (modulo isometric action of the group O(n−1)×R×Z2) with C^k−1functions z : [0, π] → Sⁿ⁻¹ such that z(0) = z(π) = (1, 0, . . . 0) and kz⁰k ≤ 1.

For a totally geodesic foliation F of Hⁿ function z is built as follows. Let 0 /∈ L ∈ F . Then z(r) is the spherical center of the subsphere L(∞) in Sⁿ⁻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) is the least upper bound kB_Lk of maxima of the quadratic form B on any unit tangent space to L.

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

kBFk = sup{kBLk | L ∈ F }.

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

(i) all the leaves of F are Hadamard manifolds

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

In codimension 1, a condition for separating leaf boundaries is an estimate of normal curvature of F by kB_Fk.

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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 ideal boundary.

Theorem 5.5 (Fenley [10]). Let M be a closed irreducible 3–manifold with non–elementary negatively curved fundamental group, F a codimension 1 Reebless 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 leaf its limit set is is not equal to S².

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

Theorem 5.6 (Fenley [11]). If a foliation F on a 3–dimensional atoroidal closed manifold M is almost transverse to a quasi-geodesic pseudo–Anosov flow then π₁(M ) is negatively curved and F has continuous extension property i.e. ideal boundary of leaf universal cover embed topologically into ideal boundary 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, Birh¨auser 1995.

[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 their ideal 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 Foliations 2005, 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 of Chicago Press 1996.

[10] S. Fenley, Limit sets of foliations in hyperbolic 3–manifolds, Topology 37 (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.

in non–positive curvature

Boundaries