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.