Ricci curvature of real hypersurfaces in non-flat complex space forms

TL;DR

This paper derives an inequality relating Ricci curvature, mean curvature, and normal curvature for real hypersurfaces in complex space forms, and classifies certain hypersurfaces satisfying the equality in non-flat cases.

Contribution

It introduces a new inequality among curvature quantities and classifies hypersurfaces satisfying the equality in two-dimensional non-flat complex space forms.

Findings

Established a curvature inequality for real hypersurfaces.

Classified hypersurfaces satisfying the equality in specific cases.

Identified geometric conditions for equality cases.

Abstract

We establish an inequality among the Ricci curvature, the squared mean curvature, and the normal curvature for real hypersurfaces in complex space forms. We classify real hypersurfaces in two-dimensional non-flat complex space forms which admit a unit vector field satisfying identically the equality case of the inequality.