The Gauss map on translational Riemannian manifolds and the topology of hypersurfaces

TL;DR

This paper introduces a new Gauss map for hypersurfaces in translational Riemannian manifolds, proves a Gauss-Bonnet theorem, and explores topological classifications of hypersurfaces in spheres based on curvature conditions.

Contribution

It defines translational Riemannian manifolds, establishes a Gauss map and curvature, and applies these to classify hypersurfaces in spheres, including new examples and reproofs of existing theorems.

Findings

A Gauss-Bonnet theorem for translational Riemannian manifolds.

Hypersurfaces with principal curvatures above a threshold are diffeomorphic to spheres.

Existence of hypersurfaces with large principal curvatures not homeomorphic to spheres.

Abstract

We introduce the notion of translational Riemannian manifolds and define a Gauss map for orientable immersed hypersurfaces lying in these ambients, an associated translational curvature and prove a Gauss-Bonnet theorem. We also use this Gauss map to prove that if $M^{n}$ is a compact, connected and oriented immersed hypersurface of the unit sphere $\mathbb{S}^{n+1}$ ($n\geq2$) contained in a geodesic ball of radius $R$ and whose principal curvatures are strictly bigger than $\tan\left( R/2 \right)$, then $M$ is diffeomorphic to $\mathbb{S}^{n}$. Additionally, we show that for any $\varepsilon\in(0,\sqrt{2}-1)$ there exists a compact, connected and oriented immersed hypersurface $M_{\varepsilon}$ of $\mathbb{S}^{n+1}$ whose principal curvatures are strictly bigger than $\varepsilon \tan \left( R/2 \right)$ but $M_{\varepsilon}$ is not homeomorphic to a sphere. Finally, using this previous result, we reobtain a theorem of Qiaoling Wang and Changyu Xia (see [4]) which asserts that if a compact and oriented hypersurface of $\mathbb{S}^{n+1}$ is contained in an open hemisphere and has nowhere zero Gauss-Kronecker curvature, then it is diffeomorphic to $\mathbb{S}^n$.