A General Inequality for Warped Product $CR$-Submanifolds of K\"ahler Manifolds

TL;DR

This paper establishes a new optimal inequality for warped product contact CR-submanifolds in certain Kahler manifolds, revealing geometric properties and correcting previous inequalities, with implications for intrinsic and extrinsic curvature relations.

Contribution

It introduces a general inequality for warped product contact CR-submanifolds using the Gauss equation, extending previous results and addressing limitations of earlier methods.

Findings

Proves $\,\mathcal{D}_T$-minimal property for these submanifolds.

Derives an optimal inequality involving scalar curvature and second fundamental form.

Corrects and generalizes inequalities previously obtained by Munteanu.

Abstract

In this paper, warped product contact $CR$-submanifolds in Sasakian, Kenmotsu and cosymplectic manifolds are shown to possess a geometric property; namely $\mathcal{D}_T$-minimal. Taking benefit from this property, an optimal general inequality for warped product contact $CR$-submanifolds is established in both Sasakian and Kenmotsu manifolds by means of the Gauss equation, we leave cosyplectic because it is an easy structure. Moreover, a rich geometry appears when the necessity and sufficiency are proved and discussed in the equality case. Applying this general inequality, the inequalities obtained by Munteanu are derived as particular cases, whereas the inequality obtained in [1] is corrected. Up to now, the method used by Chen and Munteanu can not extended for general ambient manifolds, this is because many limitations in using Codazzi equation. Hence, Our method depends on the Gauss equation. The inequality is constructed to involve an intrinsic invariant (scalar curvature) controlled by an extrinsic one (the second fundamental form), which provides an answer for Problem [1]. As further research directions, we have addressed a couple of open problems arose naturally during this work and depending on its results.