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ОбложкаVargas J.G. Differential geometry for physicists and mathematicians: moving frames and differential forms: from Euclid past Riemann. - Singapure: World scientific, 2014. - xvii, 293 p.: ill. - Bibliogr.: 277-283. - Ind.: p.285-293. - ISBN 978-981-4566-39-1
Шифр: (И/В31-V29) 02
 

Место хранения: 02 | Отделение ГПНТБ СО РАН | Новосибирск

Оглавление / Contents
 
Dedication ...................................................... v
Acknowledgements .............................................. vii
Preface ........................................................ ix

I  INTRODUCTION ................................................. 1

1  ORIENTATIONS ................................................. 3
   1.1  Selective capitalization of section titles .............. 3
   1.2  Classical in classical differential geometry ............ 4
   1.3  Intended readers of this book ........................... 5
   1.4  The foundations of physics in this BOOK ................. 6
   1.5  Mathematical VIRUSES .................................... 9
   1.6  FREQUENT MISCONCEPTIONS ................................ 11
   1.7  Prerequisite, anticipated mathematical CONCEPTS ........ 14

   II   TOOLS .................................................. 19

2  DIFFERENTIAL FORMS .......................................... 21
   2.1  Acquaintance with differential forms ................... 21
   2.2  Differentiable manifolds, pedestrianly ................. 23
   2.3  Differential 1-forms ................................... 25
   2.4  Differential γ-forms ................................... 30
   2.5  Exterior products of differential forms ................ 34
   2.6  Change of basis of differential forms .................. 35
   2.7  Differential forms and measurement ..................... 37
   2.8  Differentiable manifolds DEFINED ....................... 38
   2.9  Another definition of differentiable MANIFOLD .......... 40

3  VECTOR SPACES AND TENSOR PRODUCTS ........................... 43
   3.1  INTRODUCTION ........................................... 43
   3.2  Vector spaces (over the reals) ......................... 45
   3.3  Dual vector spaces ..................................... 47
   3.4  Euclidean vector spaces ................................ 48
        3.4.1  Definition ...................................... 48
        3.4.2  Orthonormal bases ............................... 49
        3.4.3  Reciprocal bases ................................ 50
        3.4.4  Orthogonalization ............................... 52
   3.5  Not quite right concept of VECTOR FIELD ................ 55
   3.6  Tensor products: theoretical minimum ................... 57
   3.7  Formal approach to TENSORS ............................. 58
        3.7.1  Definition of tensor space ...................... 58
        3.7.2  Transformation of components of tensors ......... 59
   3.8  Clifford algebra ....................................... 61
        3.8.1  Introduction .................................... 61
        3.8.2  Basic Clifford algebra .......................... 62
        3.8.3  The tangent Clifford algebra of 3-D Euclidean
               vector space .................................... 64
        3.8.4  The tangent Clifford algebra of spacetime ....... 65
        3.8.5  Concluding remarks .............................. 66

4  EXTERIOR DIFFERENTIATION .................................... 67
   4.1  Introduction ........................................... 67
   4.2  Disguised exterior derivative .......................... 67
   4.3  The exterior derivative ................................ 69
   4.4  Coordinate independent definition of exterior
        derivative ............................................. 70
   4.5  Stokes theorem ......................................... 71
   4.6  Differential operators in language of forms ............ 73
   4.7  The conservation law for scalar-valuedness ............. 77
   4.8  Lie Groups and their Lie algebras ...................... 79

III  TWO KLEIN GEOMETRIES ...................................... 83

5  AFFINE KLEIN GEOMETRY ....................................... 85
   5.1  Affine Space ........................................... 85
   5.2  The frame bundle of affine space ....................... 87
   5.3  The structure of affine space .......................... 89
   5.4  Curvilinear coordinates: holonomic bases ............... 91
   5.5  General vector basis fields ............................ 95
   5.6  Structure of affine space on SECTIONS .................. 97
   5.7  Differential geometry as calculus ...................... 99
   5.8  Invariance of connection differential FORMS ........... 101
   5.9  The Lie algebra of the affine group ................... 103
   5.10 The Maurer-Cartan equations ........................... 105
   5.11 HORIZONTAL DIFFERENTIAL FORMS ......................... 107

6  EUCLIDEAN KLEIN GEOMETRY ................................... 109
   6.1  Euclidean space and its frame bundle .................. 109
   6.2  Extension of Euclidean bundle to affine bundle ........ 112
   6.3  Meanings of covariance ................................ 114
   6.4  Hodge duality and star operator ....................... 116
   6.5  The Laplacian ......................................... 119
   6.6  Euclidean structure and integrability ................. 121
   6.7  The Lie algebra of the Euclidean group ................ 123
   6.8  Scalar-valued clifforms: Kähler calculus .............. 124
   6.9  Relation between algebra and geometry ................. 125

IV  CARTAN CONNECTIONS ........................................ 127

7  GENERALIZED GEOMETRY MADE SIMPLE ........................... 129
   7.1  Of connections and topology ........................... 129
   7.2  Planes ................................................ 130
        7.2.1  The Euclidean 2-plane .......................... 131
        7.2.2  Post-Klein 2-plane with Euclidean metric ....... 132
   7.3  The 2-sphere .......................................... 134
        7.3.1  The Columbus connection on the punctured
               2-sphere ....................................... 134
        7.3.2  The Levi-Civita connection on the 2-sphere ..... 136
        7.3.3  Comparison of connections on the 2-sphere ...... 137
   7.4  The 2-torus ........................................... 138
        7.4.1  Canonical connection of the 2-torus ............ 138
        7.4.2  Canonical connection of the metric of the
               2-torus ........................................ 140
   7.5  Abridged Riemann's equivalence problem ................ 140
   7.6  Use and misuse of Levi-Civita ......................... 141

8  AFFINE CONNECTIONS ......................................... 143
   8.1  Lie differentiation, INVARIANTS and vector fields ..... 143
   8.2  Affine connections and equations of structure ......... 147
   8.3  Tensoriality issues and second differentiations ....... 150
   8.4  Developments and annulment of connection .............. 153
   8.5  Interpretation of the affine curvature ................ 154
   8.6  The curvature tensor field ............................ 156
   8.7  Autoparallels ......................................... 158
   8.8  Bianchi identities .................................... 159
   8.9  Integrability and interpretation of the torsion ....... 160
   8.10 Tensor-valuedness and the conservation law ............ 161
   8.11 The zero-torsion case ................................. 164
   8.12 Horrible covariant derivatives ........................ 165
   8.13 Affine connections: rigorous APPROACH ................. 167

9  EUCLIDEAN CONNECTIONS ...................................... 171
   9.1  Metrics and the Euclidean environment ................. 171
   9.2  Euclidean structure and Bianchi IDENTITIES ............ 173
   9.3  The two pieces of a Euclidean connection .............. 177
   9.4  Affine extension of the Levi-Civita connection ........ 178
   9.5  Computation of the contorsion ......................... 179
   9.6  Levi-Civita connection by inspection .................. 180
   9.7  Stationary curves and Euclidean AUTOPARALLELS ......... 185
   9.8  Euclidean and Riemannian curvatures ................... 188
10 RIEMANNIAN SPACES AND PSEUDO-SPACES ........................ 191

   10.1 Klein geometries in greater DETAIL .................... 191
   10.2 The false spaces of Riemann ........................... 193
   10.3 Method of EQUIVALENCE ................................. 195
   10.4 Riemannian spaces ..................................... 197
   10.5 Annulment of connection at a point .................... 199
   10.6 Emergence and conservation of Einstein's tensor ....... 201
   10.7 EINSTEIN'S DIFFERENTIAL 3-FORM ........................ 202
   10.8 Einstein's 3—form: properties and equations ........... 205
   10.9 Einstein equations for Schwarzschild .................. 208

V  THE FUTURE? ................................................ 213

11 EXTENSIONS OF CARTAN ....................................... 215
   11.1 INTRODUCTION .......................................... 215
   11.2 Cartan-Finsler-CLIFTON ................................ 216
   11.3 Cartan-KALUZA-KLEIN ................................... 218
   11.4 Cartan-Clifford-KÄHLER ................................ 220
   11.5 Cartan-Kähler-Einstein-YANG-MILLS ..................... 221

12 UNDERSTAND THE PAST TO IMAGINE THE FUTURE .................. 225
   12.1 Introduction .......................................... 225
   12.2 History of some geometry-related algebra .............. 225
   12.3 History of modern calculus and differential forms ..... 227
   12.4 History of standard differential GEOMETRY ............. 229
   12.5 Emerging unification of calculus and geometry ......... 233
   12.6 Imagining the future .................................. 235

13 A BOOK OF FAREWELLS ........................................ 237
   13.1 Introduction .......................................... 237
   13.2 Farewell to vector algebra and calculus ............... 237
   13.3 Farewell to calculus of complex VARIABLE .............. 239
   13.4 Farewell to Dirac's CALCULUS .......................... 240
   13.5 Farewell to tensor calculus ........................... 242
   13.6 Farewell to auxiliary BUNDLES? ........................ 243

APPENDIX A: GEOMETRY OF CURVES AND SURFACES ................... 247
   A.l  Introduction .......................................... 247
   A.2  Surfaces in 3-D Euclidean space ....................... 248
        A.2.1  Representations of surfaces; metrics ........... 248
        A.2.2 Normal to a surface, orthonormal frames, area ... 250
        A.2.3 The equations of Gauss and Weingarten ........... 251
   A.3 Curves in 3-D Euclidean space .......................... 252
        A.3.1 Frenet's frame field and formulas ............... 252
        A.3.2 Geodesic frame fields and formulas .............. 253
   A.4  Curves on surfaces in 3-D Euclidean space ............. 254
        A.4.1  Canonical frame field of a surface ............. 254
        A.4.2  Principal and total curvatures; umbilics ....... 255
        A.4.3  Euler's, Meusnier's and Rodrigues'es theorems .. 256
        A.4.4  Levi-Civita connection induced from 3-D
               Euclidean space ................................ 256
        A.4.5  Theorema egregium and Codazzi equations ........ 257
        A.4.6  The Gauss-Bonnet formula ....................... 257
        A.4.7  Computation of the "extrinsic connection" of
               a surface ...................................... 259

APPENDIX B: "BIOGRAPHIES" ("PUBLI" GRAPHIES) ................. 261
   B.1  Elie Joseph Cartan (1869-1951) ........................ 261
        B.1.1  Introduction ................................... 261
        B.1.2  Algebra ........................................ 262
        B.1.3  Exterior differential systems .................. 263
        B.1.4  Genius even if we ignore his working on
               algebra, exterior systems proper and
               differential geometry .......................... 263
        B.l.5  Differential geometry .......................... 264
        B.1.6  Cartan the physicist ........................... 265
        B.l.7  Cartan as critic and mathematical technician ... 266
        B.l.8  Cartan as a writer ............................. 267
        B.1.9  Summary ........................................ 268
   B.2  Hermann Grassmann (1808-1877) ......................... 269
        B.2.1  Mini biography ................................. 269
        B.2.2  Multiplications galore ......................... 269
        B.2.3 Tensor and quotient algebras .................... 270
        B.2.4 Impact and historical context ................... 271

APPENDIX C: PUBLICATIONS BY THE AUTHOR ........................ 273

References .................................................... 277
Index ......................................................... 285

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