| Description |
1 online resource (xxxii, 477 pages) : illustrations |
| Contents |
1. Pseudo-Riemannian manifolds. 1.1. Symmetric bilinear forms and scalar products. 1.2. Pseudo-Riemannian manifolds. 1.3. Physical interpretations of pseudo-Riemannian manifolds. 1.4. Levi-Civita connection. 1.5. Parallel translation. 1.6. Riemann curvature tensor. 1.7. Sectional, Ricci and scalar curvatures. 1.8. Indefinite real space forms. 1.9. Lie derivative, gradient, Hessian and Laplacian. 1.10. Weyl conformal curvature tensor -- 2. Basics on pseudo-Riemannian submanifolds. 2.1. Isometric immersions. 2.2. Cartan-Janet's and Nash's embedding theorems. 2.3. Gauss' formula and second fundamental form. 2.4. Weingarten's formula and normal connection. 2.5. Shape operator of pseudo-Riemannian submanifolds. 2.6. Fundamental equations of Gauss, Codazzi and Ricci. 2.7. Fundamental theorems of submanifolds. 2.8. A reduction theorem of Erbacher-Magid. 2.9. Two basic formulas for submanifolds in E[symbol]. 2.10. Relationship between squared mean curvature and Ricci curvature. 2.11. Relationship between shape operator and Ricci curvature. 2.12. Cartan's structure equations -- 3. Special pseudo-Riemannian submanifolds. 3.1. Totally geodesic submanifolds. 3.2. Parallel submanifolds of (indefinite) real space forms. 3.3. Totally umbilical submanifolds. 3.4. Totally umbilical submanifolds of S[symbol] (1) and H[symbol] ( -1). 3.5. Pseudo-umbilical submanifolds of E[symbol]. 3.6. Pseudo-umbilical submanifolds of S[symbol] (1) and H[symbol] ( -1). 3.7. Minimal Lorentz surfaces in indefinite real space forms. 3.8. Marginally trapped surfaces and black holes. 3.9. Quasi-minimal surfaces in indefinite space forms -- 4. Warped products and twisted products. 4.1. Basics of warped products. 4.2. Curvature of warped products. 4.3. Warped product immersions. 4.4. Twisted products. 4.5. Double-twisted products and their characterization -- 5. Robertson-Walker spacetimes. 5.1. Cosmology, Robertson-Walker spacetimes and Einstein's field equations. 5.2. Basic properties of Robertson-Walker spacetimes. 5.3. Totally geodesic submanifolds of RW spacetimes. 5.4. Parallel submanifolds of RW spacetimes. 5.5. Totally umbilical submanifolds of RW spacetimes. 5.6. Hypersurfaces of constant curvature in RW spacetimes. 5.7. Realization of RW spacetimes in pseudo-Euclidean spaces |
| Summary |
The first part of this book provides a self-contained and accessible introduction to the subject in the general setting of pseudo-Riemannian manifolds and their non-degenerate submanifolds, only assuming from the reader some basic knowledge about manifold theory. A number of recent results on pseudo-Riemannian submanifolds are also included. The second part of this book is on [symbol]-invariants, which was introduced in the early 1990s by the author. The famous Nash embedding theorem published in 1956 was aimed for, in the hope that if Riemannian manifolds could be regarded as Riemannian submanifolds, this would then yield the opportunity to use extrinsic help. However, this hope had not been materialized as pointed out by M. Gromov in his 1985 article published in Asterisque. The main reason for this is the lack of control of the extrinsic invariants of the submanifolds by known intrinsic invariants. In order to overcome such difficulties, as well as to provide answers for an open question on minimal immersions, the author introduced in the early 1990s new types of Riemannian invariants, known as [symbol]-invariants, which are very different in nature from the classical Ricci and scalar curvatures. At the same time he was able to establish general optimal relations between [symbol]-invariants and the main extrinsic invariants. Since then many new results concerning these [symbol]-invariants have been obtained by many geometers. The second part of this book is to provide an extensive and comprehensive survey over this very active field of research done during the last two decades |
| Bibliography |
Includes bibliographical references (pages 439-462) and indexes |
| Notes |
English |
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Print version record |
| Subject |
Submanifolds.
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Riemannian manifolds.
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Geometry, Riemannian.
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Invariants.
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MATHEMATICS -- Geometry -- Differential.
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Geometry, Riemannian
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Invariants
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Riemannian manifolds
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Submanifolds
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| Form |
Electronic book
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| LC no. |
2011377751 |
| ISBN |
9789814329644 |
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9814329649 |
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1283234866 |
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9781283234863 |
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9786613234865 |
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6613234869 |
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