Differential Geometry Of Warped Product Manifolds And Submanifolds

Bang-Yen Chen

• Publisher: WSPC
• Year: 2017
• eBook ISBN: 9789813208940
• Topic: Mathematics
• Subtopic: Differential Geometry
• Index: Mathematics

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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 Manifolds
• Submanifolds
• Warped Product Manifolds
• Robertson-Walker Spacetimes and Schwarzschild Solution
• Contact Metric Manifolds and Submersions
• Kähler and Pseudo-Kähler Manifolds
• Slant Submanifolds
• Generic Submanifolds of Kähler Manifolds
• CR -submanifolds of Kähler Manifolds
• Warped Products in Riemannian and Kähler Manifolds
• Warped Product Submanifolds of Kähler Manifolds
• CR -warped Products in Complex Space Forms
• More on CR -warped Products in Complex Space Forms
• δ-invariants, Submersions and Warped Products
• Warped Products in Nearly Kähler Manifolds
• Warped Products in Para-Kähler Manifolds
• Warped Products in Sasakian Manifolds
• Warped Products in Affine Spaces

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--> 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 Manifold Key Features:

• Text Introduction to General Relativity assumes only a working knowledge of classical lagrangian mechanics
• Presentation then self-contained
• Mathematics and physics developed within the framework of a concrete physical problem
• Covers most of the important applications of general relativity
• Final special topics section takes reader up to a few areas of current research
• Strong emphasis on physical interpretation of all results
• An extensive set of accessible problems enhances and extends the coverage
• As a learning and teaching tool, the current book provides solutions to those problems

Chapter 1

Riemannian and Pseudo-Riemannian Manifolds

Riemannian geometry was first put forward in generality by B. Riemann in the middle of nineteenth century. Riemannian geometry, including the Euclidean geometry and the classical non-Euclidean geometries as the most special particular cases, deals with a broad range of more general geometries whose metric properties vary from point to point.

Under the impetus of Einstein’s Theory of General Relativity, the positiveness of the inner product induced from Riemannian metric was weakened to non-degeneracy. Consequently, one also has the notion of pseudo-Riemannian manifolds.

Development of Riemannian geometry resulted in a synthesis of diverse results concerning the geometry of surfaces and the behavior of geodesics on them, with techniques that can be applied to the study of differentiable manifolds of higher dimensions. It enabled Einstein’s general relativity theory, made profound impact on group theory and representation theory, as well as analysis, and spurred the development of algebraic and differential topology. Since every manifold admits a Riemannian metric, Riemannian geometry often helps to solve problems of differential topology. Most remarkably, by applying Riemannian geometry, G. Y. Perelman proved Thurston’s geometrization conjecture in 2003; consequently solved in the affirmative famous Poincaré’s conjecture posed in 1904.

1.1 Symmetric bilinear forms and scalar products

A symmetric bilinear form on a finite-dimensional real vector space V is a

-bilinear function B : V × V →

such that B(u, v) = B(v, u) for all u, v ∈ V. A symmetric bilinear form B is said to be positive definite (resp. positive semi-definite) if B(v, v) > 0 (resp. B(v, v) ≥ 0) for all v ≠ 0. Similarly, a symmetric bilinear form B is called negative definite (resp. negative semi-definite) if B(v, v) < 0 (resp. B(v, v) ≤ 0) for all v ≠ 0. B is said to be non-degenerate whenever B(u, v) = 0 for all u ∈ V implies v = 0.

Definition 1.1. The index of a symmetric bilinear form B on V is the dimension of the largest subspace W ⊂ V on which B|_W is negative definite.

Let B be a symmetric bilinear form on V . If...

Table of contents

• Cover page
• Title page
• Copyright
• Dedication
• Foreword
• Contents
• Preface
• 1. Riemannian and Pseudo-Riemannian Manifolds
• 2. Submanifolds
• 3. Warped Product Manifolds
• 4. Robertson-Walker Spacetimes and Schwarzschild Solution
• 5. Contact Metric Manifolds and Submersions
• 6. Kähler and Pseudo-Kähler Manifolds
• 7. Slant Submanifolds
• 8. Generic Submanifolds of Kähler Manifolds
• 9. CR-submanifolds of Kähler Manifolds
• 10. Warped Products in Riemannian and Kähler Manifolds
• 11. Warped Product Submanifolds of Kähler Manifolds
• 12. CR-warped Products in Complex Space Forms
• 13. More on CR-warped Products in Complex Space Forms
• 14. δ-invariants, Submersions and Warped Products
• 15. Warped Products in Nearly Kähler Manifolds
• 16. Warped Products in Para-Kähler Manifolds
• 17. Warped Products in Sasakian Manifolds
• 18. Warped Products in Affine Spaces
• Bibliography
• General Index
• Author Index