| Prologue |
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xi | |
| Preface |
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xiii | |
| Acknowledgments |
|
xv | |
| About the Authors |
|
xvii | |
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1 | (78) |
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1 | (13) |
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Differentiable structures defined on sets |
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14 | (10) |
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Differentiable functions and mappings |
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24 | (5) |
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Critical points and values |
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29 | (4) |
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Immersions, submanifolds, embeddings, and diffeomorphisms |
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33 | (10) |
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Constructing manifolds by inverse image. Implicit Map Theorem |
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43 | (5) |
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Submersions. Quotient manifolds |
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48 | (9) |
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57 | (4) |
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61 | (18) |
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Working with vector fields |
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61 | (7) |
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68 | (4) |
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72 | (3) |
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Transforming vector fields |
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75 | (4) |
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Tensor Fields And Differential Forms |
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79 | (40) |
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79 | (7) |
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Tensor and exterior algebras. Tensor fields |
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86 | (2) |
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Differential forms. Exterior product |
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88 | (9) |
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Lie derivative. Interior product |
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97 | (5) |
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Distributions and integral manifolds. Frobenius' Theorem. Differential ideals |
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102 | (8) |
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Almost symplectic manifolds |
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110 | (9) |
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119 | (18) |
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Orientable manifolds. Orientation-preserving maps |
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119 | (4) |
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Integration on chains. Stokes' Theorem I |
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123 | (2) |
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Integration on oriented manifolds. Stokes' theorem II |
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125 | (4) |
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129 | (8) |
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137 | (60) |
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Lie groups and Lie algebras |
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137 | (13) |
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Homomorphisms of Lie groups and lie algebras |
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150 | (5) |
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Lie subgroups and Lie subalgebras |
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155 | (9) |
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164 | (6) |
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The adjoint representation |
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170 | (8) |
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Lie groups of transformations |
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178 | (8) |
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186 | (11) |
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197 | (52) |
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197 | (9) |
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206 | (5) |
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211 | (14) |
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225 | (3) |
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228 | (8) |
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236 | (3) |
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239 | (10) |
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249 | (124) |
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249 | (5) |
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254 | (5) |
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259 | (6) |
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265 | (4) |
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Curvature and Ricci tensors |
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269 | (3) |
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272 | (5) |
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277 | (7) |
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Homogeneous Riemannian manifolds and Riemannian symmetric spaces |
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284 | (9) |
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Spaces of constant curvature |
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293 | (4) |
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Left-invariant metrics on Lie groups |
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297 | (8) |
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Gradient, divergence, codifferential, curl, Laplacian, and Hodge star operator on Riemannian manifolds |
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305 | (13) |
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Affine, Killing, conformal, projective, Jacobi, harmonic vector fields |
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318 | (15) |
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Submanifolds. Second fundamental form |
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333 | (6) |
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339 | (8) |
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Pseudo-Riemannian manifolds |
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347 | (26) |
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Some Definitions and Theorems |
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373 | (30) |
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373 | (6) |
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Tensor fields. Differential forms |
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379 | (4) |
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383 | (2) |
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385 | (6) |
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391 | (5) |
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396 | (7) |
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403 | (44) |
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Manual of ``Superficies'' |
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447 | (8) |
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455 | (2) |
| References |
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457 | (4) |
| List of Notations |
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461 | (4) |
| List of Figures |
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465 | (2) |
| Index |
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467 | |
