Séance Séminaire

Séminaire Gaston Darboux

The geometry of rays and the Gromov compactification of Riemannian manifolds." (joint work with F. Dal'bo and M. Peigné)

The notion of Busemann function was originally introduced by Herbert Busemann in the fifties as a tool to develop a theory of parallels on geodesic spaces (e.g. complete Riemannian manifolds). The Busemann functions captures the idea of "angle at infinity" between infinite geodesic rays, and this idea played an important role in the study of the topology complete noncompact Riemannian manifolds. In particular this notion has a special place in the geometry of Hadamard spaces (simply connected manifolds with nonpositive curvature) and in the dynamics of Kleinian groups. For a Hadamard manifold X, the Busemann functions yield a useful compactification of the space, which was originally introduced by M. Gromov; this compactification has the topology of a sphere and is easily understood in terms of rays. This nice picture breaks down for non-simply connected manifolds. The aim of the talk is to explain the main differences between the "visual" description of the Gromov compactification for Hadamard spaces and the non-simply connected case. I will give some examples of the main interesting pathologies in the non-simply connected case, such as: -- divergent rays having the same Busemann functions; -- points on the Gromov boundary which are not Busemann functions of any ray; -- discontinuity of the Busemann functions with respect to the initial conditions. I will also explain that, restricting to geometrically finite manifolds (the simplest class of non-simply connected, negatively curved manifolds) all the pathologies disappear and we recover a simple description of the Gromov boundary.