A BOUNDARY VERSION OF CARTAN–HADAMARD AND APPLICATIONS TO RIGIDITY

Abstract

The classical Cartan–Hadamard theorem asserts that a closed Riemannian manifold Mn with non-positive sectional curvature has universal cover [Formula: see text] diffeomorphic to ℝn, and a by-product of the proof is that [Formula: see text] is homeomorphic to Sn-1. We prove analogues of these two results in the case where Mn has a non-empty totally geodesic boundary. More precisely, if [Formula: see text], [Formula: see text] are two negatively curved Riemannian manifolds with non-empty totally geodesic boundary, of dimension n ≠ 5, we show that [Formula: see text] is homeomorphic to [Formula: see text]. We show that if [Formula: see text] and [Formula: see text] are a pair of non-positively curved Riemannian manifolds with totally geodesic boundary (possibly empty), then the universal covers [Formula: see text] and [Formula: see text] 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).