张量几何(第2版)

Introduction 0. Fundamental Not(at)ions 1. Sets 2. Functions 3. Physical Background Ⅰ. Real Vector Spaces 1. Spaces Subspace geometry, components 2. Maps Linearity, singularity, matrices 3. Operators Projections, eigenvMues, determinant, trace Ⅱ. Affine Spaces 1. Spaces Tangent vectors, parallelism, coordinates 2. Combinations of Points Midpoints, convexity 3. Maps Linear parts, translations, components Ⅲ. Dual Spaces 1. Contours, Co- and Contravariance, Dual Basis Ⅳ. Metric Vector Spaces 1. Metrics Basic geometry and examples, Lorentz geometry 2. Maps Isometries, orthogonal projections and complements, adjoints 3. Coordinates Orthonormal bases 4. Diagonalising Symmetric Operators Principal directions, isotropy Ⅴ. Tensors and Multilinear Forms 1. Multilinear Forms Tensor Products, Degree, Contraction, Raising Indices Ⅵ. Topological Vector Spaces 1. Continuity Metrics: topologies, homeomorphisms 2. Limits Convergence and continuity 3. The Usual Topology Continuity in finite dimensions 4. Compactness and Completeness Intermediate Value Theorem, convergence, extrema Ⅶ. Differentiation and Manifolds 1. Differentiation Derivative as local linear approxiamation 2. Manifolds Charts, maps, diffeomorphisms 3. Bundles and Fields Tangent and tensor bundles, metric tensors 4. Components Hairy Ball Theorem, transformation formulae, raising indic 5. Curves Parametrisation, length, integration 6. Vector Fields and Flows First order ordinary differential equations 7. Lie Brackets Commuting vector fields and flows Ⅷ. Connections and Covariant Differentiation 1. Curves and Tangent Vectors Representing a vector by a curve 2. Rolling Without Turning Differentiation along curves in embedded manifolds 3. Differentiating Sections Connections horizontal vectors, Christoffel symbols 4. Parallel Transport Integrating a connection 5. Torsion and Symmetry Torsion tensor of a connection 6. Metric Tensors and Connections Levi-Civita connection 7. Covariant Differentiation of Tensors Parallel transport, Ricci's Lemma, components, constancy Ⅸ. Geodesics 1. Local Characterisation Undeviating curves 2. Geodesics from a Point Completeness, exponential map, normal coordinates 3. Global Characterisation Criticality of length and energy, First Variation Formula 4. Maxima, Minima, Uniqueness Saddle points, mirages, Twins 'Paradox' 5. Geodesics in Embedded Manifolds Characterisation, examples 6. An Example of Lie Group Geometry 2x2 matrices as a pseudo-Riemannian manifold Ⅹ. Curvature 1. Flat Spaces Intrinsic description of local flatness 2. The Curvature Tensor Properties and Components 3. Curved Surfaces Ganssian curvature, Gauss-Bonnet Theorem 4. Geodesic Deviation Tidal effects in spacetime 5. Sectional Curvature Schur's Theorem, constant curvature 6. Ricci and Einstein Tensors Signs, geometry, Einstein manifolds, conservation equation 7. The Weyl Tensor Ⅺ. Special Relativity 1. Orienting Spacetimes Causality, particle histories 2. Motion in Flat Spacetime Inertial frames, momentum, rest mass, mass-energy 3. Fields Matter tensor, conservation 4. Forces No scalar potentials 5. Gravitational Red Shift and Curvature Measurement gives a curved metric tensor Ⅻ. General Relativity 1. How Geometry Governs Matter Equivalence principle, free fall 2. What Matter does to Geometry Einstein's equation, shape of spacetime 3. The Stars in Their Courses Geometry of the solar system, Schwarzschild solution 4. Farewell Particle Appendix.Existence and Smoothness of Flows 1. Completeness 2. Two Fixed Point Theorems 3. Sequences of Functions 4. Integrating Vector Quantities 5. The Main Proof 6. Inverse Function Theorem Bibliography Index of Notations Index