Interiors of Compact Contractible N-manifolds Are Hyperbolic (n 5)

The interior of every compact contractible PL n-manifold (n 5) supports a complete geodesic metric of strictly negative curvature. This provides a new family of simple examples illustrating the negative answer to a question of M. Gromov which asks whether metrically convex geodesic spaces which are topological manifolds must be homeomorphic to Euclidean spaces. The rst examples verifying the negative answer to this question were given by M. Davis and T. Januszkiewicz 11]. 0. Introduction One goal of Riemannian geometry is to use local information about a manifold to make conclusions about its global structure. A prime example is the classical Cartan-Hadamard Theorem which guarantees that every complete simply connected Riemannian manifold with non-positive sectional curvature at each point is diieomorphic to Euclidean space. The success of Riemannian geometry has inspired generalizations of its deenitions and methods to wider classes of spaces. One eeort, initiated by A. D. Aleksandrov (see 1], 2] and 3]) in the 1950's, and returned to prominence by M. Gromov in the 1980's, uses properties of triangles to extend the notion of curvature, (X), at a point x, to \geodesic spaces". These are metric spaces in which (as in complete Riemannian manifolds), the distance between two points can always be realized by a geodesic arc between them. A result of this theory which illustrates the extent to which it generalizes Riemannian geometry is the following version of the Cartan-Hadamard Theorem. (See 13] and 14]).