A Boundary Version of Cartan-hadamard and Applications to Rigidity

The classical Cartan-Hadamard theorem asserts that a closed Riemannian manifold M with non-positive sectional curvature has universal cover M̃ diffeomorphic to R, and a byproduct of the proof is that ∂∞M̃n is homeomorphic to Sn−1. We prove analogues of these two results in the case where M has a non-empty totally geodesic boundary. More precisely, if M 1 ,M n 2 are two negatively curved Riemannian manifolds with non-empty totally geodesic boundary, of dimension n 6= 5, we show that ∂∞M̃n 1 is homeomorphic to ∂∞M̃n 2 . We show that if M 1 and M n 2 are a pair of non-positively curved Riemannian manifolds with totally geodesic boundary (possibly empty), then the universal covers M̃ 1 and M̃ n 2 are diffeomorphic if and only if the universal covers have the same number of boundary components. We also show that the number of boundary components of the universal cover is either 0, 2, or ∞. As a sample application, we show that simple, thick, negatively curved P-manifolds of dimension ≥ 6 are topologically rigid. We include some straightforward consequences of topological rigidity (diagram rigidity, weak co-Hopf property, and the Nielson problem).