Vol. 46, No. 1, 1973

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Visibility manifolds

Patrick Barry Eberlein and Barrett O’Neill

Vol. 46 (1973), No. 1, 45–109

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 Visibility manifold. 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.

Mathematical Subject Classification 2000
Primary: 53C20
Received: 18 October 1971
Published: 1 May 1973
Patrick Barry Eberlein
Barrett O’Neill