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

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 ${\mathbb{S}}^{n+1}$ ($n\ge 2$) 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 $\epsilon \in (0,\sqrt{2}-1)$ there exists a compact, connected and oriented immersed hypersurface $M_{\epsilon }$ of ${\mathbb{S}}^{n+1}$ whose principal curvatures are strictly bigger than $\epsilon \tan \left(R/2\right)$ but $M_{\epsilon }$ 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 ${\mathbb{S}}^n$.