When strictly locally convex hypersurfaces are embedded

TL;DR

This paper proves embedding and topological results for strictly locally convex hypersurfaces in certain curved ambient spaces, extending classical theorems to new geometric contexts.

Contribution

It establishes Hadamard-Stoker type theorems for hypersurfaces in product spaces and Killing submersions, demonstrating embedding under principal curvature conditions.

Findings

Hypersurfaces with principal curvatures above a threshold are embedded.

The topology of such hypersurfaces is characterized.

Results extend classical convexity theorems to new ambient geometries.

Abstract

In this paper we will prove Hadamard-Stoker type theorems in the following ambient spaces: $\man ^n \times \r$, where $\man ^n $ is a $1/4-$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.