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:
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