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