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DIFFERENTIAL GEOMETRY AND LIE GROUPS
FOR PHYSICISTS
Differential geometry plays an increasingly important role in modern theoretical physics
andappliedmathematics.Thistextbookgivesanintroductiontogeometricaltopicsusefulin
theoreticalphysicsandappliedmathematics,includingmanifolds,tensorfields,differential
forms,connections,symplecticgeometry,actionsofLiegroups,bundlesandspinors.
Havingwrittenitinaninformalstyle,theauthorgivesastrongemphasisondevelopingthe
understandingofthegeneraltheorythroughmorethan1000simpleexercises,withcomplete
solutionsordetailedhints.Thebookwillpreparereadersforstudyingmoderntreatments
ofLagrangianandHamiltonianmechanics,electromagnetism,gaugefields,relativityand
gravitation.
DifferentialGeometryandLieGroupsforPhysicistsiswellsuitedforcoursesinphysics,
mathematics and engineering for advanced undergraduate or graduate students, and can
alsobeusedforactiveself-study.Therequiredmathematicalbackgroundknowledgedoes
notgobeyondthelevelofstandardintroductoryundergraduatemathematicscourses.
Maria´n Fecko, currently at the Department of Theoretical Physics, Comenius Uni-
versity,Bratislava,SlovakiaandattheInstituteofPhysics,SlovakAcademyofSciences,
Bratislava,worksonapplicationsofdifferentialgeometryinphysics.Hehasover15years’
experiencelecturingonmathematicalphysicsandteachingcoursesondifferentialgeometry
andLiegroups.
DIFFERENTIAL GEOMETRY AND LIE
GROUPS FOR PHYSICISTS
MARIA´ N FECKO
ComeniusUniversity,Bratislava,Slovakia
and
SlovakAcademyofSciences,Bratislava,Slovakia
cambridge university press
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
TheEdinburghBuilding,Cambridgecb22ru,UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Informationo nthi stitle :www.cambri dge.org/9780521845076
©M.Fecko2006
Thispublicationisincopyright.Subjecttostatutoryexceptionandtotheprovisionof
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
Firstpublishedinprintformat 2006
isbn-13 978-0-511-24521-3eBook(EBL)
isbn-10 0-511-24521-1 eBook(EBL)
isbn-13 978-0-521-84507-6hardback
isbn-10 0-521-84507-6 hardback
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Contents
Preface pagexi
Introduction 1
1 Theconceptofamanifold 4
1.1 Topologyandcontinuousmaps 4
1.2 ClassesofsmoothnessofmapsofCartesianspaces 6
1.3 Smoothstructure,smoothmanifold 7
1.4 Smoothmapsofmanifolds 11
1.5 AtechnicaldescriptionofsmoothsurfacesinRn 16
SummaryofChapter1 20
2 Vectorandtensorfields 21
2.1 Curvesandfunctionson M 22
2.2 Tangentspace,vectorsandvectorfields 23
2.3 Integralcurvesofavectorfield 30
2.4 Linearalgebraoftensors(multilinearalgebra) 34
2.5 Tensorfieldson M 45
2.6 Metrictensoronamanifold 48
SummaryofChapter2 53
3 Mappingsoftensorsinducedbymappingsofmanifolds 54
3.1 Mappingsoftensorsandtensorfields 54
3.2 Inducedmetrictensor 60
SummaryofChapter3 63
4 Liederivative 65
4.1 Localflowofavectorfield 65
4.2 LietransportandLiederivative 70
4.3 PropertiesoftheLiederivative 72
4.4 ExponentoftheLiederivative 75
4.5 Geometricalinterpretationofthecommutator[V,W],
non-holonomicframes 77
4.6 Isometriesandconformaltransformations,Killing
equations 81
SummaryofChapter4 91
v
vi Contents
5 Exterioralgebra 93
5.1 Motivation:volumesofparallelepipeds 93
5.2 p-formsandexteriorproduct 95
5.3 Exterioralgebra(cid:2)L∗ 102
5.4 Interiorproductiv 105
5.5 Orientationin L 106
5.6 DeterminantandgeneralizedKroneckersymbols 107
5.7 Themetricvolumeform 112
5.8 Hodge(duality)operator∗ 118
SummaryofChapter5 125
6 Differentialcalculusofforms 126
6.1 Formsonamanifold 126
6.2 Exteriorderivative 128
6.3 Orientability,Hodgeoperatorandvolumeformon M 133
6.4 V-valuedforms 139
SummaryofChapter6 143
7 Integralcalculusofforms 144
7.1 Quantitiesundertheintegralsignregardedas
differentialforms 144
7.2 Euclideansimplicesandchains 146
7.3 Simplicesandchainsonamanifold 149
7.4 Integralofaformoverachainonamanifold 150
7.5 Stokes’theorem 151
7.6 Integraloveradomainonanorientablemanifold 153
7.7 IntegraloveradomainonanorientableRiemannianmanifold 159
7.8 Integralandmapsofmanifolds 161
SummaryofChapter7 163
8 ParticularcasesandapplicationsofStokes’theorem 164
8.1 Elementarysituations 164
8.2 DivergenceofavectorfieldandGauss’theorem 166
8.3 CodifferentialandLaplace–deRhamoperator 171
8.4 Greenidentities 177
8.5 Vectoranalysisin E3 178
8.6 Functionsofcomplexvariables 185
SummaryofChapter8 188
9 Poincare´ lemmaandcohomologies 190
9.1 Simpleexamplesofclosednon-exactforms 191
9.2 Constructionofapotentialoncontractiblemanifolds 192
9.3∗ CohomologiesanddeRhamcomplex 198
SummaryofChapter9 203
10 Liegroups:basicfacts 204
10.1 Automorphismsofvariousstructuresandgroups 204
Contents vii
10.2 Liegroups:basicconcepts 210
SummaryofChapter10 213
11 DifferentialgeometryonLiegroups 214
11.1 Left-invarianttensorfieldsonaLiegroup 214
11.2 LiealgebraG ofagroupG 222
11.3 One-parametersubgroups 225
11.4 Exponentialmap 227
11.5 DerivedhomomorphismofLiealgebras 230
11.6 InvariantintegralonG 231
11.7 MatrixLiegroups:enjoysimplifications 232
SummaryofChapter11 243
12 RepresentationsofLiegroupsandLiealgebras 244
12.1 Basicconcepts 244
12.2 Irreducibleandequivalentrepresentations,Schur’slemma 252
12.3 Adjointrepresentation,Killing–Cartanmetric 259
12.4 Basicconstructionswithgroups,Liealgebrasandtheirrepresentations 269
12.5 Invarianttensorsandintertwiningoperators 278
12.6∗Liealgebracohomologies 282
SummaryofChapter12 287
13 ActionsofLiegroupsandLiealgebrasonmanifolds 289
13.1 Actionofagroup,orbitandstabilizer 289
13.2 Thestructureofhomogeneousspaces,G/H 294
13.3 Coveringhomomorphism,coveringsSU(2)→ SO(3)and
SL(2,C)→ L↑+ 299
13.4 RepresentationsofG andG inthespaceoffunctionsonaG-space,
fundamentalfields 310
13.5 RepresentationsofG andG inthespaceoftensorfieldsoftypeρˆ 319
SummaryofChapter13 325
14 Hamiltonianmechanicsandsymplecticmanifolds 327
14.1 Poissonandsymplecticstructureonamanifold 327
14.2 Darbouxtheorem,canonicaltransformationsandsymplectomorphisms 336
14.3 Poincare´–CartanintegralinvariantsandLiouville’stheorem 341
14.4 Symmetriesandconservationlaws 346
14.5∗Momentmap 349
14.6∗Orbitsofthecoadjointaction 354
14.7∗Symplecticreduction 360
SummaryofChapter14 368
15 Paralleltransportandlinearconnectionon M 369
15.1 Accelerationandparalleltransport 369
15.2 Paralleltransportandcovariantderivative 372
15.3 Compatibilitywithmetric,RLCconnection 382
15.4 Geodesics 389
viii Contents
15.5 Thecurvaturetensor 401
15.6 ConnectionformsandCartanstructureequations 406
15.7 Geodesicdeviationequation(Jacobi’sequation) 418
15.8∗Torsion,completeparallelismandflatconnection 422
SummaryofChapter15 428
16 Fieldtheoryandthelanguageofforms 429
16.1 DifferentialformsintheMinkowskispace E1,3 430
16.2 Maxwell’sequationsintermsofdifferentialforms 436
16.3 Gaugetransformations,actionintegral 441
16.4 Energy–momentumtensor,space-timesymmetriesandconservation
lawsduetothem 448
16.5∗Einsteingravitationalfieldequations,HilbertandCartanaction 458
16.6∗Non-linearsigmamodelsandharmonicmaps 467
SummaryofChapter16 476
17 DifferentialgeometryonTM andT∗M 478
17.1 TangentbundleTM andcotangentbundleT∗M 478
17.2 Conceptofafiberbundle 482
17.3 ThemapsTf andT∗f 485
17.4 Verticalsubspace,verticalvectors 487
17.5 LiftsonTM andT∗M 488
17.6 CanonicaltensorfieldsonTM andT∗M 494
17.7 Identitiesbetweenthetensorfieldsintroducedhere 497
SummaryofChapter17 497
18 HamiltonianandLagrangianequations 499
18.1 Second-orderdifferentialequationfields 499
18.2 Euler–Lagrangefield 500
18.3 ConnectionbetweenLagrangianandHamiltonianmechanics,
Legendremap 505
18.4 Symmetriesliftedfromthebasemanifold(configurationspace) 508
18.5 Time-dependentHamiltonian,actionintegral 518
SummaryofChapter18 522
19 Linearconnectionandtheframebundle 524
19.1 Framebundleπ : LM → M 524
19.2 Connectionformon LM 527
19.3 k-dimensionaldistributionDonamanifoldM 530
19.4 Geometricalinterpretationofaconnectionform:horizontal
distributiononLM 538
19.5 Horizontaldistributionon LM andparalleltransporton M 543
19.6 Tensorson M inthelanguageof LM andtheirparalleltransport 545
SummaryofChapter19 550
20 ConnectiononaprincipalG-bundle 551
20.1 PrincipalG-bundles 551
