Warped product contact CR-submanifolds of globally framed f-manifolds with Lorentz metric

TL;DR

This paper investigates warped product contact CR-submanifolds within indefinite S-manifolds, establishing conditions for CR-product structures and deriving geometric inequalities related to their second fundamental form.

Contribution

It introduces new conditions under which warped CR-submanifolds are CR-products and derives a geometric inequality for their second fundamental form.

Findings

Warped CR-submanifolds are CR-products under certain invariance conditions.

A geometric inequality for the second fundamental form is established.

Results apply to both spacelike and timelike cases in indefinite S-manifolds.

Abstract

In the present paper, we study globally framed f-manifolds in the particular setting of indefinite S-manifolds for both spacelike and timelike cases. We prove that if $M = N^{\perp} \times_f N^T$ is a warped CR-submanifold such that $N^{\perp}$ is $\phi$?-anti-invariant and NT is $\phi$?-invariant, then M is a CR-product. We show that the second fundamental form of a contact CR warped product of a indefinite S space form satisfies a geometric inequality, $\|h\|^2 \geq p{3\|\nabla lnf\|^2 - \triangle lnf + (c + 2)k + 1}$.