Several of the basic features
of automorphic function theory—notably the notion of limit set—can be extended to
apply to the study of Riemannian manifolds M of nonpositive curvature. Under
somewhat stronger curvature conditions e.g. K ≦ c < 0)M is called a Visibilitymanifold. For such manifolds there results a classification into three types: parabolic,axial, and fuchsian. This trichotomy is closely related to many of the most basic
topological and geometric properties of M, and such relationships will be
examined in some detail. For example, the trichotomy may be expressed in
terms of the number (suitably counted) of closed geodesics in M, namely:
0,1, or ∞. As to methodology: the conventional analytic machinery of C∞
Riemannian geometry is used, at least initially; however, at many crucial
points it will be the qualitative behavior of geodesics (ála Busemann) that is
important.
