Lectures on the geometry of manifolds

Introduction......Page 2
Space and Coordinatization......Page 12
The implicit function theorem......Page 14
Basic definitions......Page 16
Examples......Page 19
How many manifolds are there?......Page 29
Tangent spaces......Page 31
The tangent bundle......Page 34
Sard's Theorem......Page 36
Vector bundles......Page 40
Some examples of vector bundles......Page 44
Tensor products......Page 48
Symmetric and skew-symmetric tensors......Page 52
The ``super'' slang......Page 59
Duality......Page 63
Some complex linear algebra......Page 70
Operations with vector bundles......Page 74
Tensor fields......Page 76
Fiber bundles......Page 79
Flows on manifolds......Page 85
The Lie derivative......Page 87
Examples......Page 92
The exterior derivative......Page 94
Examples......Page 100
Covariant derivatives......Page 101
Parallel transport......Page 106
The curvature of a connection......Page 107
Holonomy......Page 110
The Bianchi identities......Page 113
Connections on tangent bundles......Page 114
Integration of 1-densities......Page 116
Orientability and integration of differential forms......Page 120
Stokes' formula......Page 127
Representations and characters of compact Lie groups......Page 131
Fibered calculus......Page 138
Definitions and examples......Page 142
The Levi-Civita connection......Page 145
The exponential map and normal coordinates......Page 151
The length minimizing property of geodesics......Page 153
Calculus on Riemann manifolds......Page 158
Definitions and properties......Page 168
Examples......Page 172
Cartan's moving frame method......Page 174
The geometry of submanifolds......Page 177
The Gauss-Bonnet theorem for oriented surfaces......Page 183
The 1-dimensional Euler-Lagrange equations......Page 191
Noether's conservation principle......Page 196
Variational formulæ......Page 200
Jacobi fields......Page 204
The Fundamental group and Covering Spaces......Page 211
Basic notions......Page 212
Of categories and functors......Page 216
Definitions and examples......Page 217
Unique lifting property......Page 219
Homotopy lifting property......Page 220
On the existence of lifts......Page 221
The universal cover and the fundamental group......Page 223
Speculations around the Poincaré lemma......Page 225
Cech vs. DeRham......Page 229
Very little homological algebra......Page 231
Functorial properties of the DeRham cohomology......Page 238
Some simple examples......Page 241
The Mayer-Vietoris principle......Page 243
The Künneth formula......Page 247
Cohomology with compact supports......Page 249
The Poincaré duality......Page 253
Cycles and their duals......Page 257
Intersection theory......Page 262
The topological degree......Page 267
Thom isomorphism theorem......Page 269
Gauss-Bonnet revisited......Page 272
Symmetric spaces......Page 276
Symmetry and cohomology......Page 279
The cohomology of compact Lie groups......Page 282
Invariant forms on Grassmannians and Weyl's integral formula......Page 283
The Poincaré polynomial of a complex Grassmannian......Page 290
Cech cohomology......Page 296
Sheaves and presheaves......Page 297
Cech cohomology......Page 301
Connections in principal G-bundles......Page 311
G-vector bundles......Page 317
Invariant polynomials......Page 318
The Chern-Weil Theory......Page 321
The invariants of the torus Tn......Page 325
Chern classes......Page 326
Pontryagin classes......Page 328
The Euler class......Page 330
Universal classes......Page 333
Reductions......Page 339
The Gauss-Bonnet-Chern theorem......Page 345
Co-area formulæ......Page 354
Invariant measures on linear Grassmannians......Page 365
Affine Grassmannians......Page 374
The shape operator and the second fundamental form of a submanifold in Rn......Page 377
The Gauss-Bonnet theorem for hypersurfaces of an Euclidean space.......Page 380
Gauss-Bonnet theorem for domains in an Euclidean space......Page 385
Tame geometry......Page 388
Invariants of the orthogonal group......Page 394
The tube formula and curvature measures......Page 398
Tube formula -3mu Gauss-Bonnet formula for arbitrary submanifolds......Page 408
Curvature measures of domains in an Euclidean space......Page 410
Crofton Formulæ for domains of an Euclidean space......Page 413
Crofton formulæ for submanifolds of an Euclidean space......Page 423
Basic notions......Page 430
Examples......Page 436
Formal adjoints......Page 438
Functional framework......Page 443
Sobolev spaces in RN......Page 444
Embedding theorems: integrability properties......Page 450
Embedding theorems: differentiability properties......Page 455
Functional spaces on manifolds......Page 459
Elliptic partial differential operators: analytic aspects......Page 462
Elliptic estimates in RN......Page 463
Elliptic regularity......Page 468
An application: prescribing the curvature of surfaces......Page 472
Elliptic operators on compact manifolds......Page 482
The Fredholm theory......Page 483
Spectral theory......Page 491
Hodge theory......Page 496
Basic definitions and examples......Page 500
Clifford algebras......Page 503
Clifford modules: the even case......Page 507
Clifford modules: the odd case......Page 511
A look ahead......Page 512
The spin group......Page 514
The complex spin group......Page 522
Low dimensional examples......Page 525
Dirac bundles......Page 529
The Hodge-DeRham operator......Page 533
The Hodge-Dolbeault operator......Page 538
The spin Dirac operator......Page 544
The spinc Dirac operator......Page 549