Umbilicity of constant mean curvature hypersurfaces into space forms

TL;DR

This paper identifies specific conditions on the traceless second fundamental form of complete constant mean curvature hypersurfaces in space forms, proving they are necessarily totally umbilical under these conditions.

Contribution

It establishes new criteria linking the traceless second fundamental form to the umbilicity of hypersurfaces in space forms.

Findings

Conditions on the traceless second fundamental form imply total umbilicity.

Complete constant mean curvature hypersurfaces are totally umbilical under these conditions.

Provides a geometric characterization of hypersurfaces in space forms.

Abstract

In this paper we establish conditions on the length of the traceless part of the second fundamental form of a complete constant mean curvature hypersurface immersed in a space of constant sectional curvature in order to show that it is totally umbilical.