| Introduction | p. xi |
| Pseudo-Riemannian Manifolds | p. 1 |
| Connections | p. 1 |
| First results on pseudo-Riemannian manifolds | p. 5 |
| Associate connection | p. 5 |
| Curvature | p. 6 |
| Covariant differentiation and divergence | p. 9 |
| Divergence of the Ricci tensor | p. 9 |
| Lie derivative and infinitesimal isometries | p. 11 |
| Laplacians | p. 12 |
| Sobolev spaces of tensors on Riemannian manifolds | p. 14 |
| Lorentzian manifolds | p. 16 |
| Definitions | p. 16 |
| Specific notation for Lorentzian manifolds | p. 17 |
| Introduction to Relativity | p. 19 |
| Classical fluid mechanics | p. 19 |
| A lemma on derivation of integrals | p. 19 |
| Mass of a fluid. Continuity equation | p. 20 |
| Total force | p. 21 |
| Cauchy principle | p. 22 |
| Differential expression of the motion equations | p. 22 |
| Kinematics of special relativity | p. 23 |
| Inertial Systems | p. 23 |
| Postulates of special relativity | p. 24 |
| Lorentz transformations | p. 25 |
| Inertial systems and the Minkowski space | p. 29 |
| Contraction of lengths | p. 30 |
| Proper time of a particle | p. 31 |
| Time dilation | p. 33 |
| Dynamics of special relativity | p. 34 |
| Mass and momentum | p. 34 |
| Collision laws. Equivalence of mass and energy | p. 35 |
| Minkowski force | p. 37 |
| Relativistic fluid dynamics | p. 38 |
| Stress-energy tensor of a fluid | p. 40 |
| General relativity | p. 41 |
| Fundamentals | p. 41 |
| Einstein's field equation | p. 42 |
| Cosmological models | p. 45 |
| Appendix: a theorem in affine geometry | p. 46 |
| Approximation of Einstein's Equation by the Wave Equation | p. 49 |
| Perturbations of the Ricci tensor | p. 49 |
| Einstein's equation for small perturbations of the Minkowski metric | p. 53 |
| Action on metrics of diffeomorphisms close to the identity | p. 55 |
| Continuing the calculation of Section 2 | p. 58 |
| Comparison with classical gravitation | p. 60 |
| Cauchy Problem for Einstein's Equation with Matter | p. 63 |
| Differential operators in an open set of Rn+1 | p. 64 |
| Differential operators in vector bundles | p. 70 |
| Harmonic maps | p. 73 |
| Admissible classes of stress-energy tensors | p. 76 |
| Differential operator associated to Einstein's equation | p. 79 |
| Constraint equations | p. 81 |
| Hyperbolic reduction | p. 88 |
| Fundamental theorem | p. 90 |
| An example: the stress-energy tensor of holonomic media | p. 101 |
| The Cauchy problem in the vacuum | p. 107 |
| Stability by Linearization of Einstein's Equation, General Concepts | p. 109 |
| Classical concept of stability by linearization of Einstein's equation in the vacuum | p. 109 |
| A new concept of stability by linearization of Einstein's equation in the presence of matter | p. 111 |
| How to apply the definition of stability by linearization of Einstein's equation in the presence of matter | p. 115 |
| Change of notation | p. 120 |
| Technical details concerning the map ¿ | p. 120 |
| Tangent linear map of ¿ | p. 125 |
| General Results on Stability by Linearization when the Submanifold M of V is Compact | p. 129 |
| Adjoint of D(g,k))¿ | p. 129 |
| Results | p. 133 |
| A result | p. 135 |
| Appendix: General results on elliptic operators in compact manifolds | p. 143 |
| Stability by Linearization of Einstein's Equation at Minkowski's Initial Metric | p. 149 |
| A further expression of D(g,k)¿ | p. 150 |
| The relation between Euclidean Laplacian and stability by linearization at the initial Minkowski metric | p. 152 |
| Some proofs on topological isomorphisms in Rn | p. 153 |
| Stability of the Minkowski metric: Y. Choquet-Bruhat and S. Deser's result | p. 162 |
| The Euclidean asymptotic case | p. 164 |
| Wp,s¿ (Rn) Sobolev spaces and their duals | p. 167 |
| Some results on elliptic and Fredholm operators in Rn | p. 169 |
| Proof of Theorems VII.10 and VII.11 | p. 172 |
| Stability by Linearization of Einstein's Equation in Robertson-Walker Cosmological Models | p. 177 |
| Euclidean model | p. 179 |
| Hyperbolic model | p. 180 |
| Sobolev spaces and hyperbolic Laplacian | p. 181 |
| ¿ gives an isomorphism between Fs (H3) and Fs-2(H3) | p. 182 |
| A draft of the proof of Theorem VIII.3 | p. 187 |
| Spherical model | p. 190 |
| First and second derivatives of ¿ | p. 190 |
| Adjoint map of D¿ | p. 192 |
| Proof of instability | p. 193 |
| Universes that are not simply connected | p. 198 |
| References | p. 201 |
| Index | p. 205 |
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