C-totally real warped product submanifolds

TL;DR

This paper establishes new inequalities relating the Laplacian of the warping function and the mean curvature for C-totally real warped product submanifolds in various space forms, providing obstructions to minimal immersions and an example satisfying equality.

Contribution

It introduces a general inequality for warped product submanifolds without curvature restrictions and applies it to specific space forms, also providing existence obstructions and an explicit example.

Findings

Derived a basic inequality involving Laplacian and mean curvature.

Identified obstructions to minimal isometric immersions.

Constructed an example satisfying the equality case.

Abstract

We obtain a basic inequality involving the Laplacian of the warping function and the squared mean curvature of any warped product isometrically immersed in a Riemannian manifold without assuming any restriction on the Riemann curvature tensor of the ambient manifold. Applying this general theory, we obtain basic inequalities involving the Laplacian of the warping function and the squared mean curvature of $C$-totally real warped product submanifolds of $(\kappa ,\mu ) $-space forms, Sasakian space forms and non-Sasakian $(\kappa ,\mu) $-manifolds. Then we obtain obstructions to the existence of minimal isometric immersions of $C$-totally real warped product submanifolds in $(\kappa ,\mu) $-space forms, non-Sasakian $(\kappa ,\mu) $-manifolds and Sasakian space forms. In the last, we obtain an example of a warped product $C$-totally real submanifold of a non-Sasakian $(\kappa ,\mu) $-manifold, which satisfies the equality case of the basic inequality.