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Preface to the Revised Second Edition | xi | |
Preface to the Second Edition | xiii | |
Preface to the First Edition | xv | |
I. | Introduction to Manifolds | |
1. | Preliminary Comments on R[superscript n] | 1 |
2. | R[superscript n] and Euclidean Space | 4 |
3. | Topological Manifolds | 6 |
4. | Further Examples of Manifolds. Cutting and Pasting | 11 |
5. | Abstract Manifolds. Some Examples | 14 |
II. | Functions of Several Variables and Mappings | |
1. | Differentiability for Functions of Several Variables | 20 |
2. | Differentiability of Mappings and Jacobians | 25 |
3. | The Space of Tangent Vectors at a Point of R[superscript n] | 29 |
4. | Another Definition of T[subscript a](R[superscript n]) | 32 |
5. | Vector Fields on Open Subsets of R[superscript n] | 36 |
6. | The Inverse Function Theorem | 41 |
7. | The Rank of a Mapping | 46 |
III. | Differentiable Manifolds and Submanifolds | |
1. | The Definition of a Differentiable Manifold | 52 |
2. | Further Examples | 59 |
3. | Differentiable Functions and Mappings | 65 |
4. | Rank of a Mapping, Immersions | 68 |
5. | Submanifolds | 74 |
6. | Lie Groups | 80 |
7. | The Action of a Lie Group on a Manifold. Transformation Groups | 87 |
8. | The Action of a Discrete Group on a Manifold | 93 |
9. | Covering Manifolds | 98 |
IV. | Vector Fields on a Manifold | |
1. | The Tangent Space at a Point of a Manifold | 104 |
2. | Vector Fields | 113 |
3. | One-Parameter and Local One-Parameter Groups Acting on a Manifold | 119 |
4. | The Existence Theorem for Ordinary Differential Equations | 127 |
5. | Some Examples of One-Parameter Groups Acting on a Manifold | 135 |
6. | One-Parameter Subgroups of Lie Groups | 142 |
7. | The Lie Algebra of Vector Fields on a Manifold | 146 |
8. | Frobenius's Theorem | 153 |
9. | Homogeneous Spaces | 160 |
V. | Tensors and Tensor Fields on Manifolds | |
1. | Tangent Covectors | 171 |
Covectors on Manifolds | 172 | |
Covector Fields and Mappings | 174 | |
2. | Bilinear Forms. The Riemannian Metric | 177 |
3. | Riemannian Manifolds as Metric Spaces | 181 |
4. | Partitions of Unity | 186 |
Some Applications of the Partition of Unity | 188 | |
5. | Tensor Fields | 192 |
Tensors on a Vector Space | 192 | |
Tensor Fields | 194 | |
Mappings and Covariant Tensors | 195 | |
The Symmetrizing and Alternating Transformations | 196 | |
6. | Multiplication of Tensors | 199 |
Multiplication of Tensors on a Vector Space | 199 | |
Multiplication of Tensor Fields | 201 | |
Exterior Multiplication of Alternating Tensors | 202 | |
The Exterior Algebra on Manifolds | 206 | |
7. | Orientation of Manifolds and the Volume Element | 207 |
8. | Exterior Differentiation | 212 |
An Application to Frobenius's Theorem | 216 | |
VI. | Integration on Manifolds | |
1. | Integration in R[superscript n] Domains of Integration | 223 |
Basic Properties of the Riemann Integral | 224 | |
2. | A Generalization to Manifolds | 229 |
Integration on Riemannian Manifolds | 232 | |
3. | Integration on Lie Groups | 237 |
4. | Manifolds with Boundary | 243 |
5. | Stokes's Theorem for Manifolds | 251 |
6. | Homotopy of Mappings. The Fundamental Group | 258 |
Homotopy of Paths and Loops. The Fundamental Group | 259 | |
7. | Some Applications of Differential Forms. The de Rham Groups | 265 |
The Homotopy Operator | 268 | |
8. | Some Further Applications of de Rham Groups | 272 |
The de Rham Groups of Lie Groups | 276 | |
9. | Covering Spaces and Fundamental Group | 280 |
VII. | Differentiation on Riemannian Manifolds | |
1. | Differentiation of Vector Fields along Curves in R[superscript n] | 289 |
The Geometry of Space Curves | 292 | |
Curvature of Plane Curves | 296 | |
2. | Differentiation of Vector Fields on Submanifolds of R[superscript n] | 298 |
Formulas for Covariant Derivatives | 303 | |
[down triangle, open subscript x subscript p] Y and Differentiation of Vector Fields | 305 | |
3. | Differentiation on Riemannian Manifolds | 308 |
Constant Vector Fields and Parallel Displacement | 314 | |
4. | Addenda to the Theory of Differentiation on a Manifold | 316 |
The Curvature Tensor | 316 | |
The Riemannian Connection and Exterior Differential Forms | 319 | |
5. | Geodesic Curves on Riemannian Manifolds | 321 |
6. | The Tangent Bundle and Exponential Mapping. Normal Coordinates | 326 |
7. | Some Further Properties of Geodesics | 332 |
8. | Symmetric Riemannian Manifolds | 340 |
9. | Some Examples | 346 |
VIII. | Curvature | |
1. | The Geometry of Surfaces in E[superscript 3] | 355 |
The Principal Curvatures at a Point of a Surface | 359 | |
2. | The Gaussian and Mean Curvatures of a Surface | 363 |
The Theorema Egregium of Gauss | 366 | |
3. | Basic Properties of the Riemann Curvature Tensor | 371 |
4. | Curvature Forms and the Equations of Structure | 378 |
5. | Differentiation of Covariant Tensor Fields | 384 |
6. | Manifolds of Constant Curvature | 391 |
Spaces of Positive Curvature | 394 | |
Spaces of Zero Curvature | 396 | |
Spaces of Constant Negative Curvature | 397 | |
References | 403 | |
Index | 411 |
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