  We prove a topological rigidity theorem for closed hypersurfaces of the
Euclidean sphere and of an elliptic space form. It asserts that, under a lower
bound hypothesis on the absolute value of the principal curvatures, the
hypersurface is diffeomorphic to a sphere or to a quotient of a sphere by a
group action. We also prove another topological rigidity result for
hypersurfaces of the sphere that involves the spherical image of its usual
Gauss map.
