## Semi-Riemannian Geometry With Applications to RelativityThis book is an exposition of semi-Riemannian geometry (also called pseudo-Riemannian geometry)--the study of a smooth manifold furnished with a metric tensor of arbitrary signature. The principal special cases are Riemannian geometry, where the metric is positive definite, and Lorentz geometry. For many years these two geometries have developed almost independently: Riemannian geometry reformulated in coordinate-free fashion and directed toward global problems, Lorentz geometry in classical tensor notation devoted to general relativity. More recently, this divergence has been reversed as physicists, turning increasingly toward invariant methods, have produced results of compelling mathematical interest. |

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### Contents

1 | |

34 | |

CHAPTER 3 SEMIRIEMANNIAN MANIFOLDS | 54 |

CHAPTER 4 SEMIRIEMANNIAN SUBMANIFOLDS | 97 |

CHAPTER 5 RIEMANNIAN AND LORENTZ GEOMETRY | 126 |

CHAPTER 6 SPECIAL RELATIVITY | 158 |

CHAPTER 7 CONSTRUCTIONS | 185 |

CHAPTER 8 SYMMETRY AND CONSTANT CURVATURE | 215 |

CHAPTER 11 HOMOGENEOUS AND SYMMETRIC SPACES | 300 |

CHAPTER 12 GENERAL RELATIVITY COSMOLOGY | 332 |

CHAPTER 13 SCHWARZSCHILD GEOMETRY | 364 |

CHAPTER 14 CAUSALITY IN LORENTZ MANIFOLDS | 401 |

FUNDAMENTAL GROUPS AND COVERING MANIFOLDS | 441 |

LIE GROUPS | 446 |

NEWTONIAN GRAVITATION | 453 |

456 | |

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### Common terms and phrases

achronal Cauchy hypersurface causal curve compact components conjugate points constant curvature coordinate system Corollary covariant covering map curve segment defined Definition derivative diffeomorphism endpoint equation Euclidean example Exercise finite follows formula freely falling function future-pointing G-orientation geometry gives GL(n grad hence hyperbolic hypersurface identity II(V implies inextendible integral curve isomorphism Jacobi field Killing vector field Lemma Lie algebra Lie group lightlike linear isometry Lorentz manifold matrix metric tensor Newtonian nondegenerate normal neighborhood null geodesic one-form one-to-one open set orbit orientation orthogonal orthonormal basis parallel translation piecewise smooth plane Proof Proposition radial relative restspace result Riemannian manifold scalar product Schwarzschild semi-Riemannian manifold simply connected smooth map spacelike spacetime subgroup submanifold subspace suppose symmetric space tangent space tangent vector tensor field theorem time-orientable timecone timelike curve topological totally geodesic unique unit vector variation vector space velocity warped product zero