Geometry of $CRS$ bi-warped product submanifolds in Sasakian and cosymplectic manifolds
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...
Saved in:
Main Authors | , , , , |
---|---|
Format | Journal Article |
Language | English |
Published |
07.11.2018
|
Subjects | |
Online Access | Get full text |
Cover
Loading…
Summary: | 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. |
---|---|
DOI: | 10.48550/arxiv.1811.02767 |