General inequalities and new shape operator inequality for contact CR-warped product submanifolds in cosymplectic space form

TL;DR

This paper establishes new geometric inequalities for contact CR-warped product submanifolds in cosymplectic space forms, including bounds for the second fundamental form and shape operator, with applications to nearly cosymplectic manifolds.

Contribution

It introduces novel inequalities for the shape operator and second fundamental form of contact CR-warped product submanifolds, extending results to nearly cosymplectic manifolds and involving harmonic series.

Findings

Derived inequalities for the second fundamental form and shape operator.

Identified $\,\mathcal{D}_1$-minimality as a key property.

Extended inequalities to nearly cosymplectic manifolds.

Abstract

We establish two main inequalities; one for the norm of the second fundamental form and the other for the matrix of the shape operator. The results obtained are for cosymplectic manifolds and, for these, we show that the contact warped product submanifolds naturally possess a geometric property; namely $\mathcal{D}_1$-minimality which, by means of the Gauss equation, allows us to obtain an optimal general inequality. For sake of generalization, we state our hypotheses for nearly cosymplectic manifolds, then we obtain them as particular cases for cosymplectic manifolds. For the other part of the paper, we derived some inequalities and applied them to construct and introduce a shape operator inequality for cosimpleptic manifolds involving the harmonic series. As further research directions, we have addressed a couple of open problems arose naturally during this work and which depend on its results.