Inequalities for Casorati curvatures of submanifolds in real space forms

TL;DR

This paper provides new proofs and inequalities involving Casorati curvatures of submanifolds in real space forms, characterizes a specific ideal hypersurface, and proves its rigidity.

Contribution

It introduces alternative proofs for curvature inequalities and characterizes and proves rigidity of a special Casorati ideal hypersurface.

Findings

New inequalities for normalized δ-Casorati curvature

Characterization of Casorati ideal hypersurface in Euclidean 4-space

Proof of rigidity for the hypersurface

Abstract

By using T. Oprea's optimization methods on submanifolds, we give another proof of the inequalities relating the normalized $\delta-$Casorati curvature $\hat{\delta}_c(n-1)$ for submanifolds in real space forms. Also, inequalities relating the normalized $\delta-$Casorati curvature $\delta_C(n-1)$ for submanifolds in real space forms are obtained. Besides, we characterize a kind of Casorati ideal hypersurface of Euclidean 4-space. We also show that this kind of Casorati ideal hypersurface is rigid.