Bi-warped product submanifolds of nearly Kaehler manifolds

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: $$\Vert h{\Vert }^2\ge 2p\Vert \nabla (\ln f_1){\Vert }^2+4q\left(1+\frac{10}{9}{\cot }^2\theta \right)\Vert \nabla (\ln f_2){\Vert }^2,$$ 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.

12 pages