A survey on geometry of warped product submanifolds

TL;DR

This survey reviews the development and key results concerning warped product submanifolds within various ambient manifolds, highlighting their significance in differential geometry and physics.

Contribution

It provides a comprehensive overview of the theory of warped product submanifolds, serving as an introductory reference and summarizing recent advances in the field.

Findings

Summarizes fundamental results on warped product submanifolds

Highlights their applications in geometry and physics

Identifies open problems and future research directions

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 $N_1$. The notion of warped product manifolds is one of the most fruitful generalizations of Riemannian products. Such notion plays very important roles in differential geometry as well as in physics, especially in general relativity. Warped product manifolds have been studied for a long period of time. In contrast, the study of warped product submanifolds was only initiated around the beginning of this century. In this article we survey important results on warped product submanifolds in various ambient manifolds. It is the author's hope that this survey article will provide a good introduction on the theory of warped product submanifolds as well as a useful reference for further research on this vibrant research subject.