Cargando…
Pseudo-Riemannian geometry, [delta]-invariants and applications /
Pseudo-Riemannian geometry, δ-invariants and applications /
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...
| Clasificación: | Libro Electrónico |
|---|---|
| Autor principal: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Singapore :
World Scientific,
©2011.
|
| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- 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.