| Contents |
| |
| Preface | vii |
| |
| List of Figures | xi |
| |
| 1 | Finite-Dimensional Vector Spaces and Linear Mappings | 1 |
| | 1.1 | Fields | 1 |
| | 1.2 | Finite-Dimensional Vector Spaces | 3 |
| | 1.3 | Linear Mappings of a Vector Space | 9 |
| | 1.4 | Dual or Covariant Vector Spaces | 11 |
| |
| 2 | Tensor Algebra | 16 |
| | 2.1 | Second-Order Tensors | 16 |
| | 2.2 | Higher-Order Tensors | 24 |
| | 2.3 | Exterior or Grassmann Algebra | 31 |
| | 2.4 | Inner Product Vector Spaces and the Metric Tensor | 42 |
| |
| 3 | Tensor Analysis on a Differentiate Manifold | 52 |
| | 3.1 | Differentiate Manifolds | 52 |
| | 3.2 | Tangent Vectors, Cotangent Vectors, and Parametrized Curves | 60 |
| | 3.3 | Tensor Fields over Differentiate Manifolds | 69 |
| | 3.4 | Differential Forms and Exterior Derivatives | 80 |
| |
| 4 | Differentiate Manifolds with Connections | 92 |
| | 4.1 | The Affine Connection and Covariant Derivative | 92 |
| | 4.2 | Covariant Derivatives of Tensors along a Curve | 101 |
| | 4.3 | Lie Bracket, Torsion, and Curvature Tensor | 107 |
| |
| 5 | Riemannian and Pseudo-Riemannian Manifolds | 121 |
| | 5.1 | Metric Tensor, Christoffel Symbols, and Ricci Rotation Coefficients | 121 |
| | 5.2 | Covariant Derivatives and the Curvature Tensor | 135 |
| | 5.3 | Curves, Prenet-Serret Formulas, and Geodesics | 157 |
| | 5.4 | Special Coordinate Charts | 181 |
| |
| 6 | Special Riemannian and Pseudo-Riemannian Manifolds | 200 |
| | 6.1 | Flat Manifolds | 200 |
| | 6.2 | The Space of Constant Curvature | 205 |
| | 6.3 | Einstein Spaces | 214 |
| | 6.4 | Conformally Flat Spaces | 216 |
| |
| 7 | Hypersurfaces, Submanifolds, and Extrinsic Curvature | 225 |
| | 7.1 | Two-Dimensional Surfaces Embedded in a Three-Dimensional Space | 225 |
| | 7.2 | (N - 1)-Dimensional Hypersurfaces | 233 |
| | 7.3 | D-Dimensional Submanifolds | 245 |
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| Appendix 1 Fibre Bundles | 257 |
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| Appendix 2 Lie Derivatives | 263 |
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| Answers and Hints to Selected Exercises | 271 |
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| References | 277 |
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| List of Symbols | 280 |
| |
| Index | 285 |