Chen's inequality for C-totally real submanifolds in a generalized $(\kappa ,\mu)$-space form

TL;DR

This paper establishes Chen's inequality for C-totally real submanifolds within generalized $(,)$-space forms, relating intrinsic and extrinsic curvature invariants.

Contribution

It derives new inequalities connecting scalar, sectional, Ricci, and mean curvatures for these submanifolds, extending Chen's inequality to a broader geometric context.

Findings

Derived Chen's inequality involving scalar and sectional curvatures.

Established inequalities between squared mean curvature and Ricci curvatures.

Extended curvature relations to generalized $(,)$-space forms.

Abstract

In this paper, we obtain a basic Chen's inequality for a C-totally real submanifold in a generalized $(\kappa ,\mu)$-contact space forms involving intrinsic invariants, namely the scalar curvature and the sectional curvatures of the submanifold on left hand side and the main extrinsic invariant, namely the squared mean curvature on the right hand side. Inequalities between the squared mean curvature and Ricci curvature and between the squared mean curvature and $k$-Ricci curvature are also obtained.