Pseudo-riemannian geometry, [delta]-invariants and applications / Bang-Yen Chen.

Format:

Book

Author/Creator:

Chen, Bang-yen.

Language:

English

Subjects (All):

Submanifolds.

Riemannian manifolds.

Geometry, Riemannian.

Invariants.

Physical Description:

1 online resource (510 p.)

Place of Publication:

Singapore : World Scientific, c2011.

Language Note:

English

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

Contents:

Preface; Foreword; 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; 1.3.1 4D spacetimes; 1.3.2 Kaluza-Klein theory and pseudo-Riemannian manifolds of higher dimension; 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 Submanifolds2.1 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 Ems; 2.10 Relationship between squared mean curvature and Ricci curvature; 2.11 Relationship between shape operator and Ricci curvature

2.12 Cartan's structure equations3. 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 Sm s (1) and Hm s (-1); 3.5 Pseudo-umbilical submanifolds of Ems; 3.6 Pseudo-umbilical submanifolds of Sm s (1) and Hm s (-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 products4.2 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

6. Hodge Theory, Elliptic Differential Operators and Jacobi's Elliptic Functions6.1 Operators d, . and δ; 6.2 Hodge-Laplace operator; 6.3 Elliptic differential operator; 6.4 Hodge-de Rham decomposition and its applications; 6.5 The fundamental solution of heat equation; 6.6 Spectra of some important Riemannian manifolds; 6.7 Spectra of flat tori; 6.8 Heat equation, Jacobi's elliptic and theta functions; 7. Submanifolds of Finite Type; 7.1 Order and type of submanifolds; 7.2 Minimal polynomial criterion; 7.3 A variational minimal principle; 7.4 Classification of 1-type submanifolds

7.5 Finite type immersions of compact homogeneous spaces

Notes:

Description based upon print version of record.

Includes bibliographical references and indexes.

ISBN:

9786613234865

9781283234863

1283234866

9789814329644

9814329649

OCLC:

754793931