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Semi-Riemannian Geometry With Applications to Relativity, 103.

Author: Barrett O'Neill
Publisher: Burlington : Elsevier Science, 2014.
Series: Pure and applied mathematics.
Edition/Format:   eBook : Document : EnglishView all editions and formats
Summary:
This 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  Read more...
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Genre/Form: Electronic books
Additional Physical Format: Print version:
O'Neill, Barrett.
Semi-Riemannian Geometry With Applications to Relativity, 103.
Burlington : Elsevier Science, ©2014
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Barrett O'Neill
ISBN: 9780080570570 0080570577
OCLC Number: 1058932430
Notes: Reductive Homogeneous Spaces.
Description: 1 online resource (483 pages).
Contents: Front Cover; Semi-Riemannian Geometry: With Applications to Relativity; Copyright Page; Table of Contents; Preface; Notation and Terminology; CHAPTER 1. MANIFOLD THEORY; Smooth Manifolds; Smooth Mappings; Tangent Vectors; Differential Maps; Curves; Vector Fields; One-Forms; Submanifolds; Immersions and Submersions; Topology of Manifolds; Some Special Manifolds; Integral Curves; CHAPTER 2. TENSORS; Basic Algebra; Tensor Fields; Interpretations; Tensors at a Point; Tensor Components; Contraction; Covariant Tensors; Tensor Derivations; Symmetric Bilinear Forms; Scalar Products. CHAPTER 3. SEMI-RIEMANNIAN MANIFOLDSIsometries; The Levi-Civita Connection; Parallel Translation; Geodesics; The Exponential Map; Curvature; Sectional Curvature; Semi-Riemannian Surfaces; Type-Changing and Metric Contraction; Frame Fields; Some Differential Operators; Ricci and Scalar Curvature; Semi-Riemannian Product Manifolds; Local Isometries; Levels of Structure; CHAPTER 4. SEMI-RIEMANNIAN SUBMANIFOLDS; Tangents and Normals; The Induced Connection; Geodesics in Submanifolds; Totally Geodesic Submanifolds; Semi-Riemannian Hypersurfaces; Hyperquadrics; The Codazzi Equation. Totally Umbilic HypersurfacesThe Normal Connection; A Congruence Theorem; Isometric Immersions; Two-Parameter Maps; CHAPTER 5. RIEMANNIAN AND LORENTZ GEOMETRY; The Gauss Lemma; Convex Open Sets; Arc Length; Riemannian Distance; Riemannian Completeness; Lorentz Causal Character; Timecones; Local Lorentz Geometry; Geodesics in Hyperquadrics; Geodesics in Surfaces; Completeness and Extendibility; CHAPTER 6. SPECIAL RELATIVITY; Newtonian Space and Time; Newtonian Space-Time; Minkowski Spacetime; Minkowski Geometry; Particles Observed; Some Relativistic Effects; Lorentz-Fitzgerald Contraction. Energy-MomentumCollisions; An Accelerating Observer; CHAPTER 7. CONSTRUCTIONS; Deck Transformations; Orbit Manifolds; Orientability; Semi-Riemannian Coverings; Lorentz Time-Orientability; Volume Elements; Vector Bundles; Local Isometries; Matched Coverings; Warped Products; Warped Product Geodesics; Curvature of Warped Products; Semi-Riemannian Submersions; CHAPTER 8. SYMMETRY AND CONSTANT CURVATURE; Jacobi Fields; Tidal Forces; Locally Symmetric Manifolds; Isometries of Normal Neighborhoods; Symmetric Spaces; Simply Connected Space Forms; Transvections; CHAPTER 9. ISOMETRIES. Semiorthogonal GroupsSome Isometry Groups; Time-Orientability and Space-Orientability; Linear Algebra; Space Forms; Killing Vector Fields; The Lie Algebra i(M); I(M) as Lie Group; Homogeneous Spaces; CHAPTER 10. CALCULUS OF VARIATIONS; First Variation; Second Variation; The Index Form; Conjugate Points; Local Minima and Maxima; Some Global Consequences; The Endmanifold Case; Focal Points; Applications; Variation of E; Focal Points along Null Geodesics; A Causality Theorem; CHAPTER 11. HOMOGENEOUS AND SYMMETRIC SPACES; More about Lie Groups; Bi-Invariant Metrics; Coset Manifolds.
Series Title: Pure and applied mathematics.

Abstract:

This 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 re.

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