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
|
