Warped product skew semi-invariant submanifolds of order 1 of a locally product Riemannian manifold

TL;DR

This paper introduces a new class of warped product skew semi-invariant submanifolds of order 1 in locally product Riemannian manifolds, providing conditions for their warped product structure and exploring geometric inequalities.

Contribution

It establishes necessary and sufficient conditions for these submanifolds to be warped products and analyzes the integrability of invariant distributions.

Findings

Characterization of warped product structure

Integrability conditions for invariant distributions

An inequality relating warping function and second fundamental form

Abstract

We introduce warped product skew semi-invariant submanifolds of order $1$ of a locally product Riemannian manifold. We give a necessary and sufficient condition for skew semi-invariant submanifold of order 1 to be a locally warped product. We also prove that the invariant distribution which is involved in the definition of the submanifold is integrable under some restrictions. Moreover, we find an inequality between the warping function and the squared norm of the second fundamental form for such submanifolds. Equality case is also discussed.