Pointwise Bi-Slant Submanifolds in Locally Conformal K\"ahler Manifolds Immersed as Warped Products

TL;DR

This paper characterizes when pointwise bi-slant submanifolds in locally conformal K"ahler manifolds can be immersed as warped products, extending Chen's inequality and identifying conditions on the Lee vector field and warping function.

Contribution

It provides a characterization theorem for such submanifolds as warped products, including conditions on the Lee vector field and warping function, and extends Chen's inequality to this setting.

Findings

Characterization theorem for warped product immersions

Necessary condition involving Lee vector field orthogonality

Extension of Chen's inequality with equality case analysis

Abstract

We study immersions of pointwise bi-slant submanifolds of locally conformal K\"ahler manifolds as warped products. In particular, we establish characterisation theorem for a pointwise bi-slant submanifold of a locally conformal K\"ahler manifold to be immersed as a warped product and show that a necessary condition is that the Lee vector field $B$ is orthogonal to the second factor and the warping function $\lambda$ satisfies $\text{grad}(\ln\lambda)=\frac{1}{2}B^T$, where $B^T$ denotes the tangential part of the Lee vector field. We also extend Chen's inequality for the squared length of the second fundamental form to our case and study the corresponding equality case.