Synthetic Geometry of Manifolds

Contents......Page 7
Preface......Page 11
Acknowledgements......Page 15
1.1 The number line R......Page 17
1.2 The basic infinitesimal spaces......Page 19
Coordinate-free aspects of Dk(n)......Page 21
Aspects of D......Page 23
1.3 The KL axiom scheme......Page 27
1.4 Calculus......Page 32
Directional derivative, and differentials......Page 33
Taylor polynomials......Page 35
Some rules of calculus......Page 38
1.5 Affine combinations of mutual neighbour points......Page 40
Degree calculus......Page 41
2.1 Manifolds......Page 43
Neighbours, neighbourhoods, monads......Page 45
Infinitesimal simplices and whiskers......Page 46
Neighbours and simplices in general spaces......Page 47
Affine combinations in manifolds......Page 48
The infinitesimal (simplicial and cubical) complexes......Page 51
2.2 Framings and 1-forms......Page 53
2.3 Affine connections......Page 58
Christoffel symbols......Page 60
2.4 Affine connections from framings......Page 67
2.5 Bundle connections......Page 72
2.6 Geometric distributions......Page 76
Bundle connections as geometric distributions......Page 84
1-dimensional distributions......Page 86
2.7 Jets and jet bundles......Page 87
Jet bundles......Page 90
Non-holonomous jets......Page 92
2.8 Infinitesimal simplicial and cubical complex of a manifold......Page 94
3.1 Simplicial, whisker, and cubical forms......Page 97
3.2 Coboundary/exterior derivative......Page 106
3.3 Integration of forms......Page 111
Curve integrals......Page 113
Higher surface integrals......Page 117
3.4 Uniqueness of observables......Page 120
The de Rham complex tonos (M)......Page 127
3.5 Wedge/cup product......Page 124
3.6 Involutive distributions and differential forms......Page 128
Group-valued forms......Page 130
Differential 1-forms with values in automorphism groups......Page 134
3.8 Differential forms with values in a vector bundle......Page 135
Covariant derivative......Page 136
3.9 Crossed modules and non-abelian 2-forms......Page 137
4.1 Tangent vectors and vector fields......Page 140
4.2 Addition of tangent vectors......Page 142
4.3 The log-exp bijection......Page 144
Affine connections as connections in the tangent bundle......Page 148
4.4 Tangent vectors as differential operators......Page 149
4.5 Cotangents, and the cotangent bundle......Page 151
4.6 The differential operator of a linear connection......Page 153
4.7 Classical differential forms......Page 155
Classical geometric distributions......Page 159
4.8 Differential forms with values in TM–M......Page 160
4.9 Lie bracket of vector fields......Page 162
Left-invariant vector fields on a Lie group......Page 165
Geometric distributions and vector fields......Page 166
Microcubes and marked microcubes......Page 168
5.1 Groupoids......Page 170
Graphs......Page 174
Connections......Page 176
5.3 Actions of groupoids on bundles......Page 180
5.4 Lie derivative......Page 185
5.5 Deplacements in groupoids......Page 187
The Lie algebroid of a groupoid......Page 189
5.6 Principal bundles......Page 191
5.7 Principal connections......Page 194
5.8 Holonomy of connections......Page 200
Holonomy......Page 202
Path connections......Page 203
“Curvature generates the local holonomy”......Page 207
6.1 Associative algebras......Page 209
6.2 Differential forms with values in groups......Page 212
6.3 Differential forms with values in a group bundle......Page 216
6.4 Bianchi identity in terms of covariant derivative......Page 220
6.5 Semidirect products; covariant derivative as curvature......Page 222
Affine Bianchi identity......Page 223
6.6 The Lie algebra of G......Page 226
6.7 Group-valued vs. Lie-algebra-valued forms......Page 228
6.8 Infinitesimal structure of…......Page 231
6.9 Left-invariant distributions......Page 237
6.10 Examples of enveloping algebras and enveloping algebra bundles......Page 239
7.1 Linear differential operators and their symbols......Page 241
Annular jets......Page 242
Alternative presentation of…......Page 244
7.2 Linear deplacements as differential operators......Page 247
7.3 Bundle-theoretic differential operators......Page 249
7.4 Sheaf-theoretic differential operators......Page 250
Soft vector bundles......Page 251
8.1 Pseudo-Riemannian metrics......Page 255
8.2 Geometry of symmetric affine connections......Page 259
Extending the log-exp bijection......Page 260
Point reflection, and midpoint formation......Page 261
The DL(V)-construction......Page 264
L-neighbours in M......Page 269
8.4 The Laplace operator......Page 270
Divergence of a vector field......Page 272
Laplacian in terms of divergence and gradient......Page 274
Harmonic functions......Page 275
Conformal diffeomorphisms......Page 276
A.1 Category theory......Page 277
A.2 Models; sheaf semantics......Page 279
Constructions vs. choices......Page 284
A.3 A simple topos model......Page 285
A.4 Microlinearity......Page 288
A.5 Linear algebra over local rings; Grassmannians......Page 290
Some classical manifolds......Page 294
A.6 Topology......Page 295
A.7 Polynomial maps......Page 298
A.8 The complex of singular cubes......Page 301
The chain complexes of singular cubes......Page 305
A.9 \"Nullstellensatz\" in multilinear algebra......Page 307
Bibliography......Page 309
Index......Page 314