On convexity of hypersurfaces in the hyperbolic space (Q960049)

The author discusses the following problem: When is a locally convex hypersurface in the hyperbolic space \({\mathbb{H}^n}\) globally convex, that is, when does it bound a convex set. The main result of the paper is the following: A proper locally convex embedding of a connected \((n-1)\)-manifold \(\mathcal{M}\) into \({\mathbb{H}^n}\) (\(n \geq 2\)), where the complement of the union of flat \((n-1)\)-dimensional submanifolds is connected, is the boundary of a convex body.