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In this paper, we prove a version of the classical Cartan-Hadamard theoremfor negatively curved manifolds, of dimension $n\neq 5$, with non-empty totallygeodesic 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 $\tildeM_2^n$. As a sample application, we show that simple, thick, negatively curvedP-manifolds of dimension $\geq 6$ are topologically rigid. We include somestraightforward consequences of topological rigidity (diagram rigidity, weakco-Hopf property, and Nielson realization problem).

A BOUNDARY VERSION OF CARTAN–HADAMARD AND APPLICATIONS TO RIGIDITY