Generalized inequalities on warped product submanifolds in nearly trans-Sasakian manifolds

TL;DR

This paper investigates warped product submanifolds in nearly trans-Sasakian manifolds, establishing non-existence results, geometric obstructions, and two new inequalities relating curvature and second fundamental form.

Contribution

It introduces new inequalities for the second fundamental form and characterizes conditions for warped product submanifolds in nearly trans-Sasakian manifolds.

Findings

Non-existence of certain warped product semi-slant submanifolds

Characterization and obstructions for specific warped product types

Two general inequalities relating curvature and second fundamental form

Abstract

In this paper, we study warped product submanifolds of nearly trans-Sasakian manifolds. The non-existence of the warped product semi-slant submanifolds of the type $N_\theta\times{_{f}N_T}$ is shown, whereas some characterization and new geometric obstructions are obtained for the warped products of the type $N_T\times{_{f}N_\theta}$. We establish two general inequalities for the squared norm of the second fundamental form. The first inequality generalizes derived inequalities for some contact metric manifolds [16, 18, 19, 24], while by a new technique, the second inequality is constructed to express the relation between extrinsic invariant (second fundamental form) and intrinsic invariant (scalar curvatures). The equality cases are also discussed.