GEOMETRY OF WARPED PRODUCT SUBMANIFOLDS: A SURVEY

The warped product N1 ×f N2 of two Riemannian manifolds (N1, g1) and (N2, g2) is the product manifold N1 × N2 equipped with the warped product metric g = g1 + f² g2, where f is a positive function on N1. 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 a series of articles [36, 39, 40, 43]. Since then the study of warped product submanifolds has become a very active research subject. 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.