Locally convex hypersurfaces of negatively curved spaces

Abstract:

A well-known theorem due to Hadamard states that if the second fundamental form of a compact immersed hypersurface M of Euclidean space ${E^n}(n \geqslant 3)$ is positive definite, then M is imbedded as the boundary of a convex body. There have been important generalizations of this theorem concerning hypersurfaces of ${E^n},{H^n}$, and ${S^n}$, but there seem to be no versions for hypersurfaces of spaces of variable curvature, and no proofs which generalize to these spaces. Our main result is that Hadamard’s theorem holds in any complete, simply connected Riemannian manifold of nonpositive sectional curvature.