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Jean Gallier

Notes on Differential Geometry and Lie Groups

 

 

 

E' un libro recente (2011) di ben 639 pagine sulla geometria differenziale delle varietà (manifolds) e i gruppi di Lie.

Jean Gallier è docente di computer science all'Università di Pennsylvania e in Francia ha lavorato in collaborazione col gruppo di Nicholas Ayache presso INRIA Sophia Antipolis.

Il testo è stato scritto per fornire agli studenti di corsi di computer science sulla percezione artificiale delle macchine le basi geometriche necessarie, e quindi è mantenuto ad un livello accettabilmente chiaro, e soprattutto lo sviluppo di ciascun argomento è completo e dettagliato, a partire dalle nozioni di base.

Ho utilizzato il capitolo 22 per approfondire l'argomento "esotico" delle algebre tensoriali (tensor algebras), e l'ho trovato di una chiarezza e completezza notevole (oltre 50 pagine), sempreché si abbiano le nozioni-base sui tensori come entità algebriche fornite da un buon testo di algebra astratta (es. Mac Lane-Birkhoff, Algebra, o Ciliberto, Algebra lineare, o Artin, Algebra) o di geometria superiore (es. Cavicchioli-Meschiari, Geometria, II).

Per quanto ho potuto vedere, le altre parti sono di eguale livello. La parte sui gruppi di Lie è accessibile, e sicuramente migliore del pessimo Gilmore, Lie Groups, Lie Algebras, and some of Their Applications, che viene spesso raccomandato agli studenti, e si trova sugli scaffali di qualche libreria italiana. Per non parlare delle deludentissime esposizioni di Wikipedia.

Se non si riesce a capire l'argomento in Gallier, si provi con Hall, Lie Groups, Lie Algebras and Representations, Springer-Verlag. Tuttavia, con il libro di Gallier si ha il vantaggio di avere non solo una visione teorica, ma di vedere le algebre di Lie in una delle loro principali applicazioni, relativa allo studio dei manifolds.

La versione è in pdf, frazionata in parti che rispecchiano gruppi di argomenti omogenei:

 

Capitoli 1-2

 

Capitoli 3-5

 

Capitoli 6-8

 

Capitoli 9-12

 

Capitoli 13-16

 

Capitoli 17-21

Capitolo 22

 

Ecco riportato di seguito l'indice dell'intera opera:

 

 

1 Introduction to Manifolds and Lie Groups...............................................................................   13

1.1 The Exponential Map.............................................................................................................   13

1.2 Some Classical Lie Groups..................................................................................................   23

1.3 Symmetric and Other Special Matrices............................................................................   27

1.4 Exponential of Some Complex Matrices...........................................................................   30

1.5 Hermitian and Other Special Matrices.............................................................................   33

1.6 The Lie Group SE(n) and the Lie Algebra se(n)...............................................................   34

1.7 The Derivative of a Function Between Normed Spaces..................................................   38

1.8 Manifolds, Lie Groups and Lie Algebras..........................................................................   47

2 Review of Groups and Group Actions.......................................................................................   69

2.1 Groups......................................................................................................................................   69

2.2 Group Actions and Homogeneous Spaces, I......................................................................   73

2.3 The Lorentz Groups O(n, 1), SO(n, 1) and SO0(n, 1).......................................................   90

2.4 More on O(p, q)....................................................................................................................... 102

2.5 Topological Groups............................................................................................................... 108

3 Manifolds.......................................................................................................................................... 115

3.1 Charts and Manifolds............................................................................................................. 115

3.2 Tangent Vectors, Tangent Spaces, Cotangent Spaces...................................................... 125

3.3 Tangent and Cotangent Bundles, Vector Fields............................................................... 137

3.4 Submanifolds, Immersions, Embeddings........................................................................... 144

3.5 Integral Curves, Flow, One-Parameter Groups.............................................................. 146

3.6 Partitions of Unity.................................................................................................................. 154

3.7 Manifolds With Boundary.................................................................................................... 159

3.8 Orientation of Manifolds...................................................................................................... 161

3.9 Covering Maps and Universal Covering Manifolds........................................................ 167

4 Construction of Manifolds From Gluing Data........................................................................ 173

4.1 Sets of Gluing Data for Manifolds...................................................................................... 173

4.2 Parametric Pseudo-Manifolds............................................................................................. 182

5 Lie Groups, Lie Algebra, Exponential Map.............................................................................. 185

5.1 Lie Groups and Lie Algebras............................................................................................... 185

5.2 Left and Right Invariant Vector Fields, Exponential Map............................................ 188

5.3 Homomorphisms, Lie Subgroups........................................................................................ 193

5.4 The Correspondence Lie Groups–Lie Algebras.............................................................. 197

5.5 More on the Lorentz Group SO0(n, 1)............................................................................... 198

5.6 More on the Topology of O(p, q) and SO(p, q).................................................................. 211

5.7 Universal Covering Groups................................................................................................. 214

6 The Derivative of exp and Dynkin’s Formula........................................................................... 217

6.1 The Derivative of the Exponential Map............................................................................. 217

6.2 The Product in Logarithmic Coordinates......................................................................... 219

6.3 Dynkin’s Formula................................................................................................................... 220

7 Bundles, Riemannian Metrics, Homogeneous Spaces............................................................ 223

7.1 Fibre Bundles.......................................................................................................................... 223

7.2 Vector Bundles........................................................................................................................ 239

7.3 Operations on Vector Bundles............................................................................................. 246

7.4 Metrics on Bundles, Reduction, Orientation................................................................... 250

7.5 Principal Fibre Bundles....................................................................................................... 255

7.6 Homogeneous Spaces, II........................................................................................................ 262

8 DifferentialForms 265

8.1 Differential Forms on Rn and de Rham Cohomology..................................................... 265

8.2 Differential Forms on Manifolds........................................................................................ 277

8.3 Lie Derivatives........................................................................................................................ 286

8.4 Vector-Valued Differential Forms...................................................................................... 293

8.5 Differential Forms on Lie Groups...................................................................................... 300

8.6 Volume Forms on Riemannian Manifolds and Lie Groups........................................... 305

9 Integration on Manifolds.............................................................................................................. 309

9.1 Integration in Rn.................................................................................................................... 309

9.2 Integration on Manifolds...................................................................................................... 310

9.3 Integration on Regular Domains and Stokes’ Theorem................................................. 312

9.4 Integration on Riemannian Manifolds and Lie Groups................................................. 315

10 Distributions and the Frobenius Theorem............................................................................. 321

10.1 Tangential Distributions, Involutive Distributions...................................................... 321

10.2 Frobenius Theorem.............................................................................................................. 323

10.3 Di.erential Ideals and Frobenius Theorem..................................................................... 327

10.4 A Glimpse at Foliations...................................................................................................... 330

11 Connections and Curvature in Vector Bundles..................................................................... 333

11.1 Connections in Vector Bundles and Riemannian Manifolds...................................... 333

11.2 Curvature and Curvature Form........................................................................................ 344

11.3 Parallel Transport............................................................................................................... 350

11.4 Connections Compatible with a Metric.......................................................................... 353

11.5 Duality between Vector Fields and Di.erential Forms................................................ 365

11.6 Pontrjagin Classes and Chern Classes, a Glimpse........................................................ 366

11.7 Euler Classes and The Generalized Gauss-Bonnet Theorem...................................... 374

12 Geodesics on Riemannian Manifolds....................................................................................... 379

12.1 Geodesics, Local Existence and Uniqueness.................................................................. 379

12.2 The Exponential Map........................................................................................................... 382

12.3 Complete Riemannian Manifolds, Hopf-Rinow, Cut Locus........................................ 387

12.4 The Calculus of Variations Applied to Geodesics......................................................... 392

13 Curvature in Riemannian Manifolds....................................................................................... 399

13.1 The Curvature Tensor.......................................................................................................... 399

13.2 Sectional Curvature............................................................................................................. 403

13.3 Ricci Curvature.................................................................................................................... 407

13.4 Isometries and Local Isometries....................................................................................... 410

13.5 Riemannian Covering Maps............................................................................................... 413

13.6 The Second Variation Formula and the Index Form..................................................... 415

13.7 Jacobi Fields and Conjugate Points................................................................................. 419

13.8 Convexity, Convexity Radius............................................................................................. 427

13.9 Applications of Jacobi Fields and Conjugate Points.................................................... 428

13.10 Cut Locus and Injectivity Radius: Some Properties.................................................. 433

14 Curvatures and Geodesics on Polyhedral Surfaces.............................................................. 437

15 The Laplace-Beltrami Operator and Harmonic Forms....................................................... 439

15.1 The Gradient, Hessian and Hodge . Operators.............................................................. 439

15.2 The Laplace-Beltrami and Divergence Operators........................................................ 442

15.3 Harmonic Forms, the Hodge Theorem, Poincar´e Duality......................................... 448

15.4 The Connection Laplacian and the Bochner Technique............................................... 450

16 Spherical Harmonics................................................................................................................... 457

16.1 Introduction, Spherical Harmonics on the Circle........................................................ 457

16.2 Spherical Harmonics on the 2-Sphere............................................................................. 460

16.3 The Laplace-Beltrami Operator....................................................................................... 467

16.4 Harmonic Polynomials, Spherical Harmonics and L2(Sn)......................................... 474

16.5 Spherical Functions and Representations of Lie Groups............................................ 483

16.6 Reproducing Kernel and Zonal Spherical Functions................................................... 490

16.7 More on the Gegenbauer Polynomials............................................................................ 499

16.8 The Funk-Hecke Formula................................................................................................... 502

16.9 Convolution on G/K, for a Gelfand Pair (G,K).............................................................. 505

17 Discrete Laplacians on Polyhedral Surfaces.......................................................................... 507

18 Metrics and Curvature on Lie Groups............................................................................... 509

18.1 Left (resp. Right) Invariant Metrics................................................................................ 509

18.2 Bi-Invariant Metrics........................................................................................................... 511

18.3 Connections and Curvature of Left-Invariant Metrics................................................ 516

18.4 The Killing Form................................................................................................................. 525

19 The Log-Euclidean Framework................................................................................................ 531

19.1 Introduction........................................................................................................................... 531

19.2 A Lie-Group Structure on SPD(n).................................................................................... 533

19.3 Log-Euclidean Metrics on SPD(n)................................................................................... 533

19.4 A Vector Space Structure on SPD(n)................................................................................ 537

19.5 Log-Euclidean Means.......................................................................................................... 537

19.6 Log-Euclidean Polya.ne Transformations...................................................................... 539

19.7 Fast Polyaffine Transforms................................................................................................ 542

19.8 A Log-Euclidean Framework for exp(S(n)).................................................................... 543

20 Statistics on Riemannian Manifolds........................................................................................ 547

21 Clifford Algebras, Clifford Groups, Pin and Spin................................................................ 549

21.1 Introduction: Rotations As Group Actions.................................................................... 549

21.2 Clifford Algebras.................................................................................................................. 551

21.3 Clifford Groups.................................................................................................................... 560

21.4 The Groups Pin(n) and Spin(n).......................................................................................... 566

21.5 The Groups Pin(p, q) and Spin(p, q)................................................................................. 572

21.6 Periodicity of the Clifford Algebras Clp,q..................................................................... 574

21.7 The Complex Clifford Algebras Cl(n,C).......................................................................... 578

21.8 The Groups Pin(p, q) and Spin(p, q) as double covers.................................................. 579

22 Tensor Algebras............................................................................................................................ 585

22.1 Tensors Products.................................................................................................................. 585

22.2 Bases of Tensor Products.................................................................................................... 593

22.3 Some Useful Isomorphisms for Tensor Products.......................................................... 595

22.4 Duality for Tensor Products.............................................................................................. 596

22.5 Tensor Algebras.................................................................................................................... 598

22.6 Symmetric Tensor Powers................................................................................................. 604

22.7 Bases of Symmetric Powers............................................................................................... 607

22.8 Some Useful Isomorphisms for Symmetric Powers..................................................... 609

22.9 Duality for Symmetric Powers.......................................................................................... 609

22.10 Symmetric Algebras.......................................................................................................... 611

22.11 Exterior Tensor Powers................................................................................................... 613

22.12 Bases of Exterior Powers................................................................................................. 618

22.13 Some Useful Isomorphisms for Exterior Powers....................................................... 620

22.14 Duality for Exterior Powers............................................................................................ 620

22.15 Exterior Algebras.............................................................................................................. 623

22.16 The Hodge star-Operator................................................................................................. 626

22.17 Testing Decomposability; Left and Right Hooks........................................................ 627

22.18 Vector-Valued Alternating Forms................................................................................. 634

22.19 Tensor Products of Modules over a Commmutative Ring........................................ 637

22.20 The Pfaffian Polynomial................................................................................................... 639