Geometry of warped product and CR-warped product submanifolds in Kaehler manifolds: modified version

TL;DR

This paper surveys recent advances in the geometry of warped product and CR-warped product submanifolds within Kaehler manifolds, highlighting their structures, properties, and related results.

Contribution

It provides a comprehensive overview of recent research findings on warped product and CR-warped product submanifolds in Kaehler manifolds, including new geometric insights.

Findings

Characterization of warped product submanifolds in Kaehler manifolds

Conditions for CR-warped product submanifolds to exist

Relationships between submanifold types and ambient Kaehler geometry

Abstract

The warped product $N_1\times_f N_2$ of two Riemannian manifolds $(N_1,g_1)$ and $(N_2,g_2)$ is the product manifold $N_1\times N_2$ equipped with the warped product metric $g=g_1+f^2 g_2$, where $f$ is a positive function on $N_1$. Warped products play very important roles in differential geometry as well as in physics. A submanifold $M$ of a Kaehler manifold $\tilde M$ is called a $CR$-warped product if it is a warped product $M_T\times_f N_\perp$ of a complex submanifold $M_T$ and a totally real submanifold $M_\perp$ of $\tilde M$. In this article we survey recent results on warped product and $CR$-warped product submanifolds in Kaehler manifolds. Several closely related results will also be presented.