Inequalities for the Casorati curvatures of real hypersurfaces in some Grassmannians

TL;DR

This paper establishes optimal inequalities relating scalar curvature and Casorati curvatures for real hypersurfaces in complex Grassmannians, identifying conditions for equality.

Contribution

It introduces new inequalities involving scalar and Casorati curvatures for hypersurfaces in specific Grassmannians, with conditions for equality.

Findings

Derived two optimal inequalities for hypersurfaces in Grassmannians.

Identified conditions under which the inequalities become equalities.

Extended the understanding of curvature relations in complex Grassmannian geometry.

Abstract

In this paper we obtain two types of optimal inequalities consisting of the normalized scalar curvature and the generalized normalized $\delta$-Casorati curvatures for real hypersurfaces of complex two-plane Grassmannians and complex hyperbolic two-plane Grassmannians. We also find the conditions on which the equalities hold.