## 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 | |

### Other editions - View all

Semi-Riemannian Geometry: With Applications to Relativity, Volume 103 Barrett O'Neill No preview available - 1983 |

### Common terms and phrases

algebra apply assertion assume basis called causal closed compact complete components condition consider constant contained converges coordinate system Corollary corresponding covering curvature curve defined Definition derivative diffeomorphism differential direct element energy equation equivalent example Exercise exists expression fact Figure fixed follows formula function geodesic geometry given gives grad hence holds hypersurface identity implies initial integral isometry isomorphism Killing Lemma lift linear Lorentz manifold matrix meet metric tensor natural neighborhood Newtonian normal null geodesic null vectors observer open set operator orbit orientation parallel particle particular plane positive preceding preserves projection Proof properties Proposition Prove relative result Riemannian scalar product segment semi-Riemannian manifold sequence simply connected smooth space spacelike spacetime speed starting subgroup submanifold suffices suppose symmetric tangent vector theorem timelike unique unit usual variation vector field zero