The Geometry Of Hemi-Slant Submanifolds of a Locally Product Riemannian Manifold

TL;DR

This paper investigates the geometric properties of hemi-slant submanifolds within locally product Riemannian manifolds, establishing integrability, characterization, and curvature inequalities for these submanifolds.

Contribution

It provides new conditions for hemi-slant submanifolds to be hemi-slant products and characterizes totally umbilical cases, advancing understanding of their geometric structure.

Findings

Anti-invariant distribution is integrable.

Necessary and sufficient conditions for hemi-slant product structure.

A basic inequality involving Ricci and mean curvature.

Abstract

In the present paper, we study hemi-slant submanifolds of a locally product Riemannian manifold. We prove that the anti-invariant distribution which is involved in the definition of hemi-slant submanifold is integrable and give some applications of this result. We get a necessary and sufficient condition for a proper hemi-slant submanifold to be a hemi-slant product. We also study this type submanifolds with parallel canonical structures. Moreover, we give two characterization theorems for the totally umbilical proper hemi-slant submanifolds. Finally, we obtain a basic inequality involving Ricci curvature and the squared mean curvature of a hemi-slant submanifold of a certain type locally product Riemannian manifold.