Topological rigidity for closed hypersurfaces of elliptic space forms

TL;DR

This paper establishes topological rigidity results for closed hypersurfaces in spheres and elliptic space forms, showing they are diffeomorphic to spheres or their quotients under certain curvature conditions.

Contribution

It introduces new topological rigidity theorems for hypersurfaces in elliptic space forms based on curvature bounds and Gauss map properties.

Findings

Hypersurfaces with curvature bounds are diffeomorphic to spheres or quotients.

New rigidity results involving the spherical image of the Gauss map.

Theorems apply to hypersurfaces in Euclidean spheres and elliptic space forms.

Abstract

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.