Differential Geometry of Warped Product Manifolds and Submanifolds

Abstract

The warped product N 1 × f N 2 of two Riemannian manifolds (N 1 , g 1 ) and (N 2 , g 2 ) is the product manifold N 1 × 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 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.

Euclidean Submanifolds via Tangential Components of Their Position Vector Fields

“…The first author proved in [34] that a Lorentzian manifold is a GRW spacetime if and only if it admits a time-like concircular vector field. For further results in this respect, see an excellent survey on GRW spacetimes by C. A. Mantica and L. G. Molinari [35] (see also [5]). …”

“…According to [7], a submanifold M of a Euclidean m-space E m is called a rectifying submanifold if the position vector field x of M always lies in its rectifying space. In other words, M is called a rectifying submanifold if and only if:…”

“…A non-trivial vector field Z on a Riemannian manifold M is called a concircular vector field if it satisfies (cf., e.g., [5,31,32]):…”

“…For instance, it was proven in [34] that a Lorentzian manifold is a generalized Robertson-Walker spacetime if and only if it admits a timelike concircular vector field. For the most recent surveys on generalized Robertson-Walker spacetimes, see [5,35].…”

Rectifying curves and geodesics on a cone in the Euclidean 3-space

“…In view of these basic facts, the author asked the following very simple natural geometric question in [3] (see also [6] We simply call such a curve a rectifying curve in [3]. It is known that rectifying curves have many interesting properties (see, for instance, [3,4,5,6,7]). …”