Abstract
Warped products are the most natural and fruitful generalization of Riemannian products. Such products play very important roles in differential geometry and in general relativity. After Bishop and O’Neill’s 1969 article, there have been many works done on warped products from intrinsic point of view during the last fifty years. In contrast, the study of warped products from extrinsic point of view was initiated around the beginning of this century by the first author in a series of his articles. In particular, he established an optimal inequality for an isometric immersion of a warped product N1×fN2 into any Riemannian manifold Rm(c) of constant sectional curvature c which involves the Laplacian of the warping function f and the squared mean curvature H2. Since then, the study of warped product submanifolds became an active research subject, and many papers have been published by various geometers. The purpose of this article is to provide a comprehensive survey on the study of warped product submanifolds which are closely related with this inequality, done during the last two decades.
Highlights
For two given Riemannian manifolds, B and F, of positive dimensions, endowed withRiemannian metrics, gB and g F, respectively, and, for a positive smooth function, f on B, the warped product N = B × f F is, by definition, the manifold B × F equipped with the warped product Riemannian metric g = gB + f 2 g F
The best relativistic model of the Schwarzschild spacetime that describes the out space around a massive star or a black hole can be described as a warped product. (For recent surveys on warped products as Riemannian submanifolds, we refer to Reference [2,4])
One of the most fundamental problems in the theory of submanifolds is the immersibility of a Riemannian manifold into a Euclidean m-space Em (or more generally, into a real space form Rm (c) of constant sectional curvature c)
Summary
For two given Riemannian manifolds, B and F, of positive dimensions, endowed with. Riemannian metrics, gB and g F , respectively, and, for a positive smooth function, f on B, the warped product N = B × f F is, by definition, the manifold B × F equipped with the warped product Riemannian metric g = gB + f 2 g F (see Reference [1]). Since Nash’s embedding theorem implies that every warped product N1 × f N2 can always be regarded as a Riemannian submanifold in some Euclidean space, a special case of the research program posted in Reference [6] is to study the two following fundamental problems: Problem 1. In the beginning of this century, the first author provided several solutions to these two fundamental problems in a series of his articles (see Reference [6,7,8,9,10]) He established in Reference [6,10] some sharp relationships between the Laplacian of the warping function and the squared mean curvature of warped product submanifolds. The main purpose of this article is to provide a comprehensive survey on the study of warped product submanifolds which are closely related with this inequality mentioned in abstract, which have been done during the last two decades
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