# Bi-warped product submanifolds of nearly Kaehler manifolds

## TL;DR

This paper investigates bi-warped product submanifolds within nearly Kaehler manifolds, establishing a sharp inequality involving the second fundamental form and the warping functions, and explores conditions for equality.

## Contribution

It introduces a new inequality for bi-warped product submanifolds in nearly Kaehler manifolds and analyzes the equality case, extending the understanding of their geometric properties.

## Key findings

- Established a sharp inequality relating second fundamental form and warping functions.
- Derived conditions under which the inequality becomes an equality.
- Provided applications of the inequality in geometric analysis.

## Abstract

We study bi-warped product submanifolds of nearly Kaehler manifolds which are the natural extension of warped products. We prove that every bi-warped product submanifold of the form $M=M_T\times_{f_1}\! M_\perp\times_{f_2}\! M_\theta$ in a nearly Kaehler manifold satisfies the following sharp inequality: $$\|h\|^2\geq 2p\|\nabla (\ln f_1)\|^2+4q\left(1+{\small \frac{10}{9}}\cot^2\theta\right)\|\nabla(\ln f_2)\|^2,$$ where $p=\dim M_\perp$, $q=\frac{1}{2}\dim M_\theta$, and $f_1,\,f_2$ are smooth positive functions on $M_T$. We also investigate the equality case of this inequality. Further, some applications of this inequality are also given.