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 $M=N_T{\times }_{f_1}{N}_{\perp }^{n_1}{\times }_{f_2}{N}_{\theta }^{n_2}$ in a cosymplectic manifold $\tilde{M}$, then its second fundamental form $h$ satisfies the inequality: $$\Vert h{\Vert }^2\ge 2n_1\Vert \nabla (\ln f_1){\Vert }^2+2n_2(1+2{\cot }^2\theta )\Vert \nabla (\ln f_2){\Vert }^2,$$ where $N_T,N_{\perp }^{n_1}$ and $N_{\theta }^{n_2}$ are invariant, anti-invariant and proper pointwise slant submanifolds of $\tilde{M}$, 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.

16 pages