Geometry of $CRS$ bi-warped product submanifolds in Sasakian and cosymplectic manifolds

Abstract

In this paper, we prove that there are no proper $CRS$ bi-warped product submanifolds other than contact CR-biwarped products in Sasakian manifolds. On the other hand, we prove that if $M$ is a $CRS$ bi-warped product of the form $M=N_T \times_{f_1}N^{n_{1}}_\perp\times_{f_2} N^{n_{2}}_\theta$ in a cosymplectic manifold $\widetilde M$, then its second fundamental form $h$ satisfies the inequality: $$\|h\|^2\geq 2n_1\|\nabla(\ln f_1)\|^2+2n_2(1+2\cot^2\theta)\|\nabla(\ln f_2)\|^2,$$ where $N_T,\, N^{n_{1}}_\perp$ and $N^{n_{2}}_\theta$ are invariant, anti-invariant and proper pointwise slant submanifolds of $\widetilde M$, respectively, and $\nabla(\ln f_1)$ and $\nabla(\ln f_2)$ denote the gradients of $\ln f_{1}$ and $\ln f_{2}$, respectively. Several applications of this inequality are given. At the end, we provide a non-trivial example of bi-warped products satisfying the equality case.