Strasbourg master class on geometry

Preface......Page 5
Contents......Page 7
Notes on non-Euclidean geometry by Norbert A’Campo and Athanase Papadopoulos......Page 9
 Introduction......Page 10
  Basic notions......Page 17
  Hilbert's axioms of neutral geometry......Page 21
  Equivalent forms of Euclid's parallel postulate......Page 26
  The axiom of hyperbolic geometry......Page 31
  Spherical geometry......Page 32
  Euclidean trigonometric formulae obtained as limits of hyperbolic and spherical trigonometric formulae......Page 37
  Comments on references......Page 39
  Some results in neutral geometry......Page 41
  Saccheri's Theorem and other results in neutral geometry......Page 45
  Angular deficit in neutral geometry......Page 49
  Trirectangular quadrilaterals in neutral geometry......Page 55
  Khayyam–Saccheri quadrilaterals......Page 58
  Projection in neutral geometry......Page 62
  Some basic properties in hyperbolic geometry......Page 63
  On quadrilaterals in hyperbolic geometry......Page 66
  Trirectangular quadrilaterals in hyperbolic geometry......Page 67
  Equidistance......Page 70
  Geometric relations in quadrilaterals......Page 74
  Introduction......Page 82
  Angular deficit in hyperbolic geometry......Page 84
  The area function......Page 86
  Dissection in Euclidean geometry......Page 90
  Dissection in non-Euclidean geometry......Page 95
  Introduction......Page 97
  The function E(y)......Page 99
  The functional equation (x+y)+(x-y)=2(x)(y)......Page 103
  Pythagoras' theorem......Page 105
  Trigonometry in an arbitrary triangle......Page 110
  Geometric relations in trirectangular quadrilaterals......Page 116
  Some spherical trigonometry......Page 118
  The function E(y) in spherical geometry......Page 121
  Introduction......Page 123
  Parallelism in hyperbolic geometry......Page 124
  Parabolic motions and horocycles......Page 133
  Horocycle contraction and applications......Page 139
  The functional equation f(x)f(y)=f(x+y)......Page 142
  Lobachevsky's angle of parallelism function......Page 143
  A Euclidean model of the hyperbolic plane......Page 147
  A model arising from algebra......Page 158
  Introduction......Page 171
  A coherent model for the three geometries......Page 173
 References......Page 181
Crossroads between hyperbolic geometry and number theory byFrançoise Dal’Bo......Page 191
 Introduction......Page 192
  Basic concepts in Riemannian geometry......Page 195
  Poincaré half-plane and Möbius transformations......Page 197
  Area, length and distance......Page 199
  Circles, neighborhoods and perpendicular bisectors......Page 202
  Boundary at infinity and projective action......Page 203
  Classification of elements of G......Page 204
  Horocycles......Page 205
  Vector approach......Page 208
 Actions of the modular group......Page 210
  The action of  on H()......Page 211
  A tiling of H......Page 212
  Hyperbolic characterization of rational numbers......Page 214
  An application to the linear action of `39`42`"613A``45`47`"603ASL(2,Z)......Page 216
 Topology of the horocyclic trajectories......Page 218
  Basic concepts in topological dynamics......Page 219
  Dynamics of U on `39`42`"613A``45`47`"603ASL(2,Z)"026E30F `39`42`"613A``45`47`"603ASL(2,R)......Page 222
  Dynamics of the horocyclic flow......Page 223
  Introduction to the geodesic flow......Page 226
  Characterization of some trajectories......Page 228
 Geodesic trajectories and Diophantine approximation......Page 231
  Classical results in number theory......Page 232
  Dynamical characterization of badly approximated numbers......Page 233
  Application to small values of binary quadratic forms......Page 235
 References......Page 238
 Motivation......Page 241
  Combinatorial definition......Page 242
  Coverings of the punctured torus......Page 243
  Monodromy......Page 244
  Subgroups of F_2......Page 245
  Translation structures......Page 246
  Variation of the translation structure......Page 247
  Teichmüller disks......Page 249
  Teichmüller and moduli space......Page 250
  The affine group......Page 251
  Veech groups of origamis......Page 253
  Characteristic origamis......Page 254
  Congruence groups......Page 255
 An example: the origami W......Page 256
 References......Page 260
Five lectures on 3-manifold topology byPhilipp Korablev and Sergey Matveev......Page 263
  Triangulations......Page 264
  Heegaard splittings......Page 265
  Surgery presentation......Page 268
  Spine presentation......Page 270
  Construction......Page 273
  One example of the Turaev–Viro invariant......Page 274
 JSJ-decomposition......Page 276
  Normalization procedure......Page 277
  Matching system......Page 279
  Fundamental surfaces......Page 281
  Introduction......Page 284
  Hierarchies and extension moves......Page 285
  Proof of Theorem 15......Page 288
 References......Page 291
An introduction to globally symmetric spaces byGabriele Link......Page 293
 Generalities on symmetric spaces......Page 294
  Geometric definition......Page 295
  The group of isometries......Page 296
  Algebraic point of view......Page 297
  Geodesics and curvature......Page 299
  Examples......Page 300
  The Killing form......Page 304
  Decomposition of symmetric spaces......Page 307
 Symmetric spaces of non-compact type......Page 311
  Flats and rank......Page 312
  Roots and root spaces......Page 316
  Iwasawa decomposition......Page 320
  The space of maximal flats......Page 321
  Weyl group and opposition involution......Page 322
  Cartan decomposition and Cartan vector......Page 323
  The geometric boundary of S......Page 325
  The Furstenberg boundary......Page 330
  The Bruhat decomposition......Page 331
  Visibility at infinity......Page 334
  Busemann functions and distances......Page 337
 References......Page 339
 References......Page 377
  Generalities on SU(2)......Page 342
  Basic examples......Page 343
  Some representation spaces of knot complements......Page 345
 Differentiable structure and twisted cohomology......Page 348
  Two examples......Page 349
  The general case......Page 350
  Applications......Page 351
  Reidemeister torsion......Page 354
  Principal bundles and flat connections......Page 356
  Sections and connection forms......Page 357
  de Rham cohomology and isomorphisms......Page 359
 Chern–Simons theory......Page 360
  The Chern–Simons functional......Page 361
  Construction of the prequantum bundle......Page 362
  Examples......Page 363
  Trace functions and flat connection along a curve......Page 364
  Global description of the moduli space......Page 366
  Some applications......Page 367
  Spin structures......Page 368
  Lagrangian foliations......Page 370
  Bohr–Sommerfeld leaves......Page 371
  Going further......Page 376
 3-dimensional ``objects''......Page 379
 Hyperbolic structures......Page 385
 Cusped manifolds......Page 391
 Complexity and closed manifolds......Page 401
 Geodesic boundary and graphs......Page 404
 References......Page 409
 Introduction......Page 413
  Basic notation......Page 418
  Motivation from geometric group theory......Page 419
  Stability of quasi-geodesics......Page 421
  -inequality and Gromov product......Page 423
  Cross difference and cross difference triple......Page 426
  Gromov product on the boundary......Page 428
  Busemann functions......Page 430
 Geodesic hyperbolic spaces......Page 432
  Geodesic boundary......Page 434
 A metric structure on _X......Page 435
 Morphisms of hyperbolic spaces......Page 436
 Möbius geometry of _X......Page 440
 Hyperbolic approximation......Page 441
 Quasimetric spaces and their Möbius geometry......Page 448
  Quasi-metrics and metrics......Page 449
  Complete quasi-metric spaces and the metric involution......Page 450
  Möbius geometry of quasi-metric spaces......Page 452
  Morphisms of quasi-metric spaces......Page 453
 Appendix: Proof of Theorem 5.6......Page 455
 References......Page 460