Differential Geometry Of Warped Product Manifolds And Submanifolds

A warped product manifold is a Riemannian or pseudo-Riemannian manifold whose metric tensor can be decomposed into a Cartesian product of the y geometry and the x geometry — except that the x-part is warped, that is, it is rescaled by a scalar function of the other coordinates y. The notion of warped product manifolds plays very important roles not only in geometry but also in mathematical physics, especially in general relativity. In fact, many basic solutions of the Einstein field equations, including the Schwarzschild solution and the Robertson–Walker models, are warped product manifolds.

The first part of this volume provides a self-contained and accessible introduction to the important subject of pseudo-Riemannian manifolds and submanifolds. The second part presents a detailed and up-to-date account on important results of warped product manifolds, including several important spacetimes such as Robertson–Walker's and Schwarzschild's.

The famous John Nash's embedding theorem published in 1956 implies that every warped product manifold can be realized as a warped product submanifold in a suitable Euclidean space. The study of warped product submanifolds in various important ambient spaces from an extrinsic point of view was initiated by the author around the beginning of this century.

The last part of this volume contains an extensive and comprehensive survey of numerous important results on the geometry of warped product submanifolds done during this century by many geometers.
Contents:Riemannian and Pseudo-Riemannian ManifoldsSubmanifoldsWarped Product ManifoldsRobertson-Walker Spacetimes and Schwarzschild SolutionContact Metric Manifolds and SubmersionsKähler and Pseudo-Kähler ManifoldsSlant SubmanifoldsGeneric Submanifolds of Kähler ManifoldsCR-submanifolds of Kähler ManifoldsWarped Products in Riemannian and Kähler ManifoldsWarped Product Submanifolds of Kähler ManifoldsCR-warped Products in Complex Space FormsMore on CR-warped Products in Complex Space Formsδ-invariants, Submersions and Warped ProductsWarped Products in Nearly Kähler ManifoldsWarped Products in Para-Kähler ManifoldsWarped Products in Sasakian ManifoldsWarped Products in Affine Spaces
Readership: Graduate students and researchers interested in warped product manifolds and submanifolds in geometry, mathematical physics and general relativity.
Warped Product Manifold;Warped Product Submanifold;Spacetime;Kaehler Manifold;Sasakian Manifold;Affine Space;Robertson–Walker Spacetime;Black Hole;Pseudo-Riemannian Manifold;Semi-Riemannian Manifold;Submanifold;Slant Submanifolds;CR-Submanifold;CR-Warped Product;Lagrangian Submanifold;Delta-Invariant;Nearly Kaehler Manifold;Para-Kaehler ManifoldKey Features:Text Introduction to General Relativity assumes only a working knowledge of classical lagrangian mechanicsPresentation then self-containedMathematics and physics developed within the framework of a concrete physical problemCovers most of the important applications of general relativityFinal special topics section takes reader up to a few areas of current researchStrong emphasis on physical interpretation of all resultsAn extensive set of accessible problems enhances and extends the coverageAs a learning and teaching tool, the current book provides solutions to those problems