When strictly locally convex hypersurfaces are embedded

In this paper we will prove Hadamard-Stoker type theorems in the following ambient spaces: \man ^n \times \r, where ${\text{man}}^n$ is a $1\mathrm{/}4-$pinched manifold, and certain Killing submersions, e.g., Berger spheres and Heisenberg spaces. That is, under the condition that the principal curvatures of an immersed hypersurfaces are greater than some non-negative constant (depending on the ambient space), we prove that such a hypersurface is embedded and we also study its topology.