I. On the interior geometry of metric spaces.- 1. Preliminaries.- 2. The Hopf-Rinow Theorem.- 3. Spaces with curvature bounded from above.- 4. The Hadamard-Cartan Theorem.- 5. Hadamard spaces.- II. The boundary at infinity.- 1. Closure of X via Busemann functions.- 2. Closure of X via rays.- 3. Classification of isometries.- 4. The cone at infinity and the Tits metric.- III. Weak hyperbolicity.- 1. The duality condition.- 2. Geodesic flows on Hadamard spaces.- 3. The flat half plane condition.- 4. Harmonic functions and random walks on ?.- IV. Rank rigidity.- 1. Preliminaries on geodesic flows.- 2. Jacobi fields and curvature.- 3. Busemann functions and horospheres.- 4. Rank, regular vectors and flats.- 5. An invariant set at infinity.- 6. Proof of the rank rigidity.- Appendix. Ergodicity of geodesic flows.- 1. Introductory remarks.- Measure and ergodic theory preliminaries.- Absolutely continuous foliations.- Anosov flows and the Ho continuity of invariant distributions.- Proof of absolute continuity and ergodicity.

Homogeneity and bounded isometries in manifolds of negative curvature

Lectures on Spaces of Nonpositive Curvature

If a group Γ is generated by a finite subset 5, then one has the "growth function" gs, where gs(m) is the number of distinct elements of Γ expressible as words of length <m on 5. Roughly speaking, J. Milnor [9] shows that the asymptotic behaviour of gs does not depend on choice of finite generating set S c Γ, and that lower (resp. upper) bounds on the curvature of a riemannian manifold M result in upper (resp. lower) bounds on the growth function of π^M). The types of bounds on the growth function are

/pdf/growth-of-finitely-generated-solvable-groups-and-curvature-w9jrd76nz8.pdf

Growth of finitely generated solvable groups and curvature of Riemannian manifolds

background

On homogeneous manifolds of negative curvature

Noncompact homogeneous einstein spaces

Let M be a Hadamard manifold, i.e., a connected, simply connected, complete riemannian manifold of nonpositive curvature. To be more precise, assume that the sectional curvature K of M satisfies — b<K< — a, where 0 < a < oo and 0 < & < o o . I f p e M and z is a point at infinity (cf. EberleinO'Neill [4], which we give as a general reference for Hadamard manifolds), there exists a horosphere through p with center z. This is defined as follows: Denote the geodesic ray from p to z by γ, and consider the geodesic spheres through p with center γ(t), t > 0. As t goes to infinity, these spheres converge to the horosphere. More precisely, the horospheres are the level surfaces of the Busemann function F = limFt, where Ft is defined by Ft(p) = d(p, γ(t)) — t. In the flat case (a = b = 0), horospheres are just affine hyperplanes, and in the case of constant negative curvature, using the Poincare model we see that horospheres are euclidean spheres internally tangent to the boundary sphere, minus the point of tangency. The main purpose of this paper is to show that, to a certain extent, the geometry of horospheres in M may be compared with that in the spaces of constant curvature — a and — b, respectively. We give two examples: 1. (Theorem 4.6). // jf is a horosphere and h denotes the distance in Jf with respect to the induced metric, then for all p,q e j f

/pdf/geometry-of-horospheres-4h1hrpfgbw.pdf

Geometry of horospheres

https://projecteuclid.org/download/pdf_1/euclid.ijm/1256067453

Homogeneity and bounded isometries in manifolds of negative curvature

Citations

Lectures on Spaces of Nonpositive Curvature

Growth of finitely generated solvable groups and curvature of Riemannian manifolds

On homogeneous manifolds of negative curvature

Noncompact homogeneous einstein spaces

Geometry of horospheres

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