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 metricg = g1 + f² g2, where f is a positive function on n1. the notion of warped productmanifolds is one of the most fruitful generalizations of riemannian products. such notionplays very important roles in differential geometry as well as in physics, especially ingeneral relativity. warped product manifolds have been studied for a long period oftime. in contrast, the study of warped product submanifolds was only initiated aroundthe beginning of this century in a series of articles [36, 39, 40, 43]. since then the studyof warped product submanifolds has become a very active research subject.in this article we survey important results on warped product submanifolds in variousambient manifolds. it is the author’s hope that this survey article will provide a goodintroduction on the theory of warped product submanifolds as well as a useful referencefor further research on this vibrant research subject.