- Title Pages
- I Lorentz Geometry
- II Special Relativity
- III General Relativity and Einstein's Equations
- IV Schwarzschild Spacetime and Black Holes
- V Cosmology
- VI Local Cauchy Problem
- VII Constraints
- VIII Other Hyperbolic-Elliptic Well-Posed Systems
- IX Relativistic Fluids
- X Relativistic Kinetic Theory
- XI Progressive Waves
- XII Global Hyperbolicity and Causality
- XIII Singularities
- XIV Stationary Spacetimes and Black Holes
- XV Global Existence Theorems: Asymptotically Euclidean Data
- XVI Global Existence Theorems: The Cosmological Case
- APPENDIX I Sobolev Spaces on Riemannian Manifolds
- APPENDIX II Second-Order Elliptic Systems on Riemannian Manifolds
- Appendix III Quasi-Diagonal, Quasi-Linear, Second-Order Hyperbolic Systems
- APPENDIX IV General Hyperbolic Systems
- APPENDIX V Cauchy–Kovalevski and Fuchs Theorems
- APPENDIX VI Conformal Methods
- APPENDIX VII Kaluza–Klein Theories
- Related Papers
- Causality of Classical Supergravity Yvonne Choquet-Bruhat I.M.T.A, Université Paris 6.
- Gravitation with Gauss Bonnet Terms Yvonne Choquet-Bruhat Abstract We study general properties of the partial differential equations for generalized gravity arising from the Lovelock Lagrangian.
- Interaction of Gravitational and Fluid Waves. Yvonne Choquet-Bruhat and Antonio Greco
- Course 6 Positive-Energy Theorems
- (p.1) I Lorentz Geometry
- General Relativity and the Einstein Equations
- Oxford University Press
This chapter presents a survey of the basic definitions of Riemannian and Lorentzian differential geometry used in this book. The first nine sections use the simplest formulations, in local coordinates, as they are needed for the first five chapters and physical applications. The later sections contain material used in the following, more advanced, chapters. Topics covered include manifolds, differential mappings, vectors and tensors, pseudo-Riemannian metrics, Riemannian connection, geodesics, curvature, geodesic deviation, maximum length and conjugate points, linearized Ricci and Einstein tensors, and second derivative of the Ricci tensor.
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