A boundary version of Cartan-Hadamard and applications to rigidity

Abstract

In this paper, we prove a version of the classical Cartan-Hadamard theorem for negatively curved manifolds, of dimension $n\neq 5$, with non-empty totally geodesic boundary. More precisely, if $M_1^n,M_2^n$ are any two such manifolds, we show that (1) $\partial ^\infty \tilde M_1^n$ is homeomorphic to $\partial ^\infty \tilde M_2^n$, and (2) $\tilde M_1^n$ is homeomorphic to $\tilde M_2^n$. As a sample application, we show that simple, thick, negatively curved P-manifolds of dimension $\geq 6$ are topologically rigid. We include some straightforward consequences of topological rigidity (diagram rigidity, weak co-Hopf property, and Nielson realization problem).