Table of Contents

Preface vii

1 Manifolds 1

1.1 Preliminaries 1

1.1.1 Space and Coordinatization 1

1.1.2 The implicit function theorem 3

1.2 Smooth manifolds 6

1.2.1 Basic definitions 6

1.2.2 Partitions of unity 9

1.2.3 Examples 10

1.2.4 How many manifolds are there? 20

2 Natural Constructions on Manifolds 23

2.1 The tangent bundle 23

2.1.1 Tangent spaces 23

2.1.2 The tangent bundle 27

2.1.3 Transversality 29

2.1.4 Vector bundles 33

2.1.5 Some examples of vector bundles 37

2.2 A linear algebra interlude 41

2.2.1 Tensor products 41

2.2.2 Symmetric and skew-symmetric tensors 46

2.2.3 The "super" slang 53

2.2.4 Duality 56

2.2.5 Some complex linear algebra 64

2.3 Tensor fields 67

2.3.1 Operations with vector bundles 67

2.3.2 Tensor fields 69

2.3.3 Fiber bundles 73

3 Calculus on Manifolds 79

3.1 The Lie derivative 79

3.1.1 Flows on manifolds 79

3.1.2 The Lie derivative 81

3.1.3 Examples 86

3.2 Derivations of Ω•(M) 88

3.2.1 The exterior derivative 88

3.2.2 Examples 93

3.3 Connections on vector bundles 94

3.3.1 Covariant derivatives 94

3.3.2 Parallel transport 100

3.3.3 The curvature of a connection 101

3.3.4 Holonomy 105

3.3.5 The Bianchi identities 108

3.3.6 Connections on tangent bundles 109

3.4 Integration on manifolds 111

3.4.1 Integration of 1-densities 111

3.4.2 Orientability and integration of differential forms 115

3.4.3 Stokes' formula 123

3.4.4 Representations and characters of compact Lie groups 127

3.4.5 Fibered calculus 133

4 Riemannian Geometry 139

4.1 Metric properties 139

4.1.1 Definitions and examples 139

4.1.2 The Levi-Civita connection 143

4.1.3 The exponential map and normal coordinates 147

4.1.4 The length minimizing property of geodesies 150

4.1.5 Calculus on Riemann manifolds 155

4.2 The Riemann curvature 165

4.2.1 Definitions and properties 165

4.2.2 Examples 169

4.2.3 Cartan's moving frame method 172

4.2.4 The geometry of submanifolds 175

4.2.5 Correlators and their geometry 182

4.2.6 The Gauss-Bonnet theorem for oriented surfaces 192

5 Elements of the Calculus of Variations 201

5.1 The least action principle 201

5.1.1 The 1-dimensional Euler-Lagrange equations 201

5.1.2 Noether's conservation principle 207

5.2 The variational theory of geodesics 210

5.2.1 Variational formulæ 211

5.2.2 Jacobi fields 214

5.2.3 The Hamilton-Jacobi equations 220

6 The Fundamental Group and Covering Spaces 227

6.1 The fundamental group 228

6.1.1 Basic notions 228

6.1.2 Of categories and functors 232

6.2 Covering Spaces 233

6.2.1 Definitions and examples 233

6.2.2 Unique lifting property 235

6.2.3 Homotopy lifting property 236

6.2.4 On the existence of lifts 237

6.2.5 The universal cover and the fundamental group 239

7 Cohomology 241

7.1 DeRham cohomology 241

7.1.1 Speculations around the Poincaré lemma 241

7.1.2 Cech vs. DeRham 245

7.1.3 Very little homological algebra 247

7.1.4 Funetorial properties of the DeRham cohomology 254

7.1.5 Some simple examples 257

7.1.6 The Mayer-Vietoris principle 259

7.1.7 The Künneth formula 262

7.2 The Poincaré duality 265

7.2.1 Cohomology with compact supports 265

7.2.2 The Poincaré duality 268

7.3 Intersection theory 272

7.3.1 Cycles and their duals 272

7.3.2 Intersection theory 277

7.3.3 The topological degree 282

7.3.4 The Thorn isomorphism theorem 284

7.3.5 Gauss-Bonnet revisited 287

7.4 Symmetry and topology 291

7.4.1 Symmetric spaces 291

7.4.2 Symmetry and cohomology 294

7.4.3 The cohomology of compact Lie groups 298

7.4.4 Invariant forms on Grassmannians and Weyl's integral formula 299

7.4.5 The Poincaré polynomial of a complex Grassmannian 306

7.5 Cech cohomology 312

7.5.1 Sheaves and presheaves 313

7.5.2 Cech cohomology 317

8 Characteristic Classes 329

8.1 Chern-Weil theory 329

8.1.1 Connections in principal G-bundles 329

8.1.2 G-vector bundles 335

8.1.3 Invariant polynomials 336

8.1.4 The Chern-Weil Theory 339

8.2 Important examples 343

8.2.1 The invariants of the torus Tⁿ 343

8.2.2 Chern classes 343

8.2.3 Pontryagin classes 346

8.2.4 The Euler class 348

8.2.5 Universal classes 351

8.3 Computing characteristic classes 357

8.3.1 Reductions 358

8.3.2 The Gauss-Bonnet-Chern theorem 363

9 Classical Integral Geometry 373

9.1 The integral geometry of real Grassmannians 373

9.1.1 Co-area formula? 373

9.1.2 Invariant measures on linear Grassmannians 386

9.1.3 Affine Grassmannians 395

9.2 Gauss-Bonnet again?!? 398

9.2.1 The shape operator and the second fundamental form 398

9.2.2 The Gauss-Bonnet theorem for hypersurfaces of an Euclidean space 401

9.2.3 Gauss-Bonnet theorem for domains in an Euclidean space 405

9.3 Curvature measures 409

9.3.1 Tame geometry 409

9.3.2 Invariants of the orthogonal group 414

9.3.3 The tube formula and curvature measures 418

9.3.4 Tube formula → Gauss-Bonnet formula for arbitrary submanifolds of an Euclidean space 429

9.3.5 Curvature measures of domains in an Euclidean space 431

9.3.6 Crofton formulæ for domains of an Euclidean space 433

9.3.7 Crofton formulæ for submanifolds of an Euclidean space 443

10 Elliptic Equations on Manifolds 451

10.1 Partial differential operators: algebraic aspects 451

10.1.1 Basic notions 451

10.1.2 Examples 457

10.1.3 Formal adjoints 459

10.2 Functional framework 464

10.2.1 Sobolev spaces in R^N 464

10.2.2 Embedding theorems: integrability properties 471

10.2.3 Embedding theorems: differentiability properties 476

10.2.4 Functional spaces on manifolds 480

10.3 Elliptic partial differential operators: analytic aspects 484

10.3.1 Elliptic estimates in R^N 485

10.3.2 Elliptic regularity 489

10.3.3 An application: prescribing the curvature of surfaces 494

10.4 Elliptic operators on compact manifolds 504

10.4.1 Fredholm theory 504

10.4.2 Spectral theory 513

10.4.3 Hodge theory 518

11 Spectral Geometry 523

11.1 Generalized functions and currents 523

11.1.1 Generalized functions and operations withy them 523

11.1.2 Currents 528

11.1.3 Temperate distributions and the Fourier transform 529

11.1.4 Linear differential equations with distributional data 531

11.2 Important families of generalized functions 536

11.2.1 Some classical generalized functions on the real axis 536

11.2.2 Homogeneous generalized functions 542

11.3 The wave equation 548

11.3.1 Fundamental solutions of the wave 548

11.3.2 The wave family 550

11.3.3 Local parametrices for the wave equation with variable coefficients 567

11.4 Spectral geometry 574

11.4.1 The spectral function of the Laplacian on a compact manifold 574

11.4.2 Short time asymptotics for the wave kernel 580

11.4.3 Spectral function asymptotics 588

11.4.4 Spectral estimates of smoothing operators 593

11.4.5 Spectral perestroika 598

12 Dirac Operators 611

12.1 The structure of Dirac operators 611

12.1.1 Basic definitions and examples 611

12.1.2 Clifford algebras 614

12.1.3 Clifford modules: the even case 617

12.1.4 Clifford modules: the odd case 622

12.1.5 A look ahead 623

12.1.6 The spin group 625

12.1.7 The complex spin group 633

12.1.8 Low dimensional examples 635

12.1.9 Dirac bundles 640

12.2 Fundamental examples 644

12.2.1 The Hodge-DeRham operator 644

12.2.2 The Hodge-Dolbeault operator 648

12.2.3 The spin Dirac operator 654

12.2.4 The spin^c Dirac operator 660

Bibliography 669

Index 675