Differential Geometry and Mathematical Physics: Part I. Manifolds, Lie Groups and Hamiltonian Systems

765 Pages·2013·10.371 MB·English

by  Rudolph, G. and Schmidt, M.

Differential Geometry and Mathematical Physics
Theoretical and Mathematical Physics
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EditorialBoard
W.Beiglböck,InstituteofAppliedMathematics,UniversityofHeidelberg,Heidelberg,
Germany
P.Chrusciel,GravitationalPhysics,UniversityofVienna,Vienna,Austria
J.-P. Eckmann, Département de Physique Théorique, Université de Genève, Geneva,
Switzerland
H.Grosse,InstituteofTheoreticalPhysics,UniversityofVienna,Vienna,Austria
A.Kupiainen,DepartmentofMathematics,UniversityofHelsinki,Helsinki,Finland
H. Löwen, Institute of Theoretical Physics, Heinrich-Heine-University of Düsseldorf,
Düsseldorf,Germany
M.Loss,SchoolofMathematics,GeorgiaInstituteofTechnology,Atlanta,USA
N.A.Nekrasov,IHÉS,Bures-sur-Yvette,France
M.Ohya,TokyoUniversityofScience,Noda,Japan
M. Salmhofer, Institute of Theoretical Physics, University of Heidelberg, Heidelberg,
Germany
S.Smirnov,MathematicsSection,UniversityofGeneva,Geneva,Switzerland
L.Takhtajan,DepartmentofMathematics,StonyBrookUniversity,StonyBrook,USA
J.Yngvason,InstituteofTheoreticalPhysics,UniversityofVienna,Vienna,Austria
Forfurthervolumes:
www.springer.com/series/720
Gerd Rudolph (cid:2) Matthias Schmidt
Differential
Geometry and
Mathematical
Physics
Part I. Manifolds, Lie Groups and
Hamiltonian Systems
GerdRudolph MatthiasSchmidt
InstituteforTheoreticalPhysics InstituteforTheoreticalPhysics
UniversityofLeipzig UniversityofLeipzig
Leipzig,Germany Leipzig,Germany
ISSN1864-5879 ISSN1864-5887(electronic)
TheoreticalandMathematicalPhysics
ISBN978-94-007-5344-0 ISBN978-94-007-5345-7(eBook)
DOI10.1007/978-94-007-5345-7
SpringerDordrechtHeidelbergNewYorkLondon
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Introduction
This book is the first of two volumes on differential geometry and mathematical
physics. It is the result of our teaching these subjects at the University of Leipzig
over the last few decades to students of physics and of mathematics. The present
volumeisdevotedtomanifolds,LiegroupsandthetheoryofHamiltoniansystems.
The second volume will deal with fibre bundles, topology and gauge field theory,
includingaspectsofthetheoryofgravity.
Whiletheaimandscopeofdifferentialgeometryaresomewhatwelldefined,it
is,perhaps,lessclearwhatwepossiblymeanbymathematicalphysics.Historically,
thistermwasratherimpreciseandsoisstillnowadays.Indeed,itsinterpretationde-
pendsoncultureandcontext.Inourunderstanding,mathematicalphysicsisthearea
where theoretical physics and pure mathematics meet, stimulate and fertilize each
other.Ontheonehand,thisinterplayleadstoadeeperstructuralunderstandingof
theoreticalphysicsandtonewresultsobtainedwithnewmathematicalmethods.On
theotherhand,itstimulatesthedevelopmentofoldandnewbranchesinmathemat-
ics.Thus,inourunderstanding,itisimpossibletodrawapreciseborderlinebetween
theoreticalphysicsandpuremathematics.Overthelastdecadesitsometimeshap-
penedthatthesolutionofaproblemposedbyphysicistshadanevenlargerimpact
onthedevelopmentofmathematicsthanonthefieldofphysicsfromwhereitarose.
Thereisanumberoftextswherethestatusandtheroleofmathematicalphysicsis
discussed,seee.g.thepapersofGreenberg[112],Faddeev[87]andJaffeandQuinn
[151],1 aswellastheclassicalcontributionsofPoincaré[241]andHilbert[128].
There is no doubt that, over the last few centuries, the interrelation between
physics and geometry has been especially tight and fruitful. In particular, this in-
teraction has stimulated the development of modern differential geometry. In this
complex process, which we cannot describe here,2 the development of the notion
of manifoldwas of great importance.The conceptualdefinitionof this notionwas
1Thisveryinterestingandsomewhatprovocativearticlestimulatedalotofresponsesbyleading
scientists,see[27].
2See[264–266]and[281,282]foradetaileddiscussionofthesehistoricalaspects.
v
vi Introduction
presented by Riemann in his famous Habilitationsvortrag in Göttingen in the year
1854, see [251].3 Riemann developed a general geometry (nowadays called Rie-
manniangeometry),whichincludedEuclideanaswellasnon-Euclideangeometry
asoriginatedbyBolyai,GaußandLobachevsky.FromRiemann’spresentationitis
clearthatadeeperunderstandingofthenatureofphysicalspacewasoneofthemain
motivationsforhisstudies.Inhisunderstanding,spaceworksonbodiesandbodies
haveaninfluenceonspace.ThisideamadeRiemanndepartfromthemetaphysical
attitudetowardsspaceasagivenunchangeableentityandpasstoamodernfieldthe-
oreticalpointofview.Heevensuggestedthatthemetricmightbedeterminedbythe
physicalmasses.Thus,onaratherphilosophicallevel,hemadeasteptowardsthe
conceptual foundations of Einstein’s theory of gravity, which came 60 years later.
Atthesametime,hecreatedthemathematicalframeworkforthistheory.
Inthefollowingyears,anumberofgreatmathematicianscontributedtothefield,
butitwasPoincaréwhobroughttheconceptofmanifoldtoitsmodernform.4Ashe
saidhimself,hewasledtothisconceptbyhispreviousinvestigationsonthetheory
ofdifferentialequationsanditsapplicationstodynamics,inparticular,ofthen-body
problem. On the one hand, on the basis of this abstract manifold concept, he laid
thefoundationsofmodernalgebraictopology.Ontheotherhand,hecontinuedhis
studiesondynamicalsystemswithemphasisontheirglobal,qualitativebehaviour,
onthewaycreatingalotoftoolswhichnowadaysstillplayanimportantrole.Atthe
sametime,heprovidedageometrizationoftheformalismofanalyticalmechanics
as developed by Lagrange,5 Hamilton,6 Jacobi, Liouville and Poisson. Instead of
formulatingdynamicsintermsoflocalcoordinatesinEuclideanspace,heviewed
it as a global system described by a Hamiltonian vector field on the phase space
manifold.Thus,modernsymplecticgeometrywasborn.7
Inthetwentiethcentury,theinteractionbetweenphysicsandgeometrycontinued
tobestrongandsuccessful.Ofcourse,firstofall,weshouldmentionEinstein’sthe-
oryofgravity,whichisbasedonthediscoverythatgravityisageometricpropertyof
spacetimeandthatspacetimeiscurvedbymatter.Startingfromthenineteen-fifties,
allotherfundamentalforcesweregeometrizedinasimilarspiritleadingtomodern
gaugetheory.Thenecessarymathematicalfoundations,includingthegeneraltheory
3Therewere,ofcourse,precursors.Inparticular,CauchyandGaußshouldbementioned.Gauß
evenusedthetermmanifold,butinhisunderstandingitwasrestrictedtoaffinesubspacesofan
n-dimensionalvectorspace.
4SeethefamousAnalysissitus[239]fromtheyear1895,togetherwithfivesubsequentcomple-
mentswithinthefollowingnineyears.Actually,Poincarégavetwodefinitionsofamanifold:a
manifoldasalevelsetandamanifoldasgivenbyanatlasoflocalcharts.
5TheoriginofsymplecticgeometrydatesbacktoLagrange’searlyworkoncelestialmechanics,
see[177]and[308]foradetaileddiscussionbyWeinstein.
6See[120–122].Hamiltonwasledtotheformulationofdynamicsintermsofasystemoffirst
orderdifferentialequationsforageneralmechanicalsystemthroughhisstudiesinoptics.
7ThewordsymplecticwasinventedbyWeyl[312]togiveanametothegroupoflineartrans-
formations,preservinganon-degenerate,skew-symmetricbilinearform,andthetermsymplectic
geometrywasproposedbySiegel,see[272].
Introduction vii
offibrebundlesandconnections,weredevelopedbyÉ.Cartan,Koszul,Ehresmann
andChern.Oncethesegeometricalformulationsofthefundamentalforceshadbeen
found, another fascinating interaction of geometry and physics took place. In the
mid-seventies physicists insisted on classifying the solutions (of a certain type) of
the Yang-Mills field equations of classical gauge theory. This problem was finally
solvedbyleadingmathematiciansandthetechniquesdevelopedontheway,inturn,
ledtofundamentallynewanddeepinsightsintopology.Inparticular,exotictopo-
logical structures on Euclidean four-space were found. We should also add that,
startingfromtheworkofKaluza,Klein,EinsteinandWeylinthenineteen-twenties
upuntilthepresent,therehasbeenmucheffortdevotedtosearchingforanultimate
geometricalmodelunifyingallfundamentalforces.Thisgaveanotherstrongimpe-
tustothedevelopmentofmoderndifferentialgeometryandrelatedfields.Someof
theaspectsjustmentionedwillbediscussedinvolume2ofthisbook.
Tofinishthisbriefhistoricalintroduction,weshouldmaketwofurtherremarks.
Firstly, it should be mentioned that in the process described above the concept of
symmetryplayedafundamentalrole.Thecreationofthemathematicalfoundations
ofthisconceptdatesbacktoLie,whointheeighteen-seventiesdevelopedageneral
theoryoftransformations.8LiewasinfluencedbytheworkofGaloisonsymmetries
ofpolynomialequations,bytheworkofJacobionpartialdifferentialequations,and
byKlein,whoseaimwas tounifygeometryandgrouptheory.Thetheorywas es-
sentially pushed forward by Killing and É. Cartan, who classified semisimple Lie
algebras and developed their representation theory. The early period closes with
thecontributionsbyWeyl,whocreatedtherepresentationtheoryofsemisimpleLie
groups. It was also Weyl who first applied concepts of group theory to quantum
mechanics.Itgoeswithoutsayingthatthegeneraltheoryoffibrebundlesandcon-
nections and, consequently, also the theory of gauge fields, heavily rests on Lie
grouptheory.
Secondly,overthelastfewdecades,ithasbecomemoreandmoreclearthatsym-
plectic geometry plays a special role. This is not only due to the fact that there is
merely a lot of applications of symplectic techniques in many areas of mathemat-
ics and in physics. There is something more: a phenomenon which Arnold called
symplectization,see[22],[111]andalso[308].Indeed,thereseemstobegrowing
evidence that many concepts, constructions and results from different branches of
mathematicsandmathematicalphysics(likethetheoryofpartialdifferentialequa-
tions,thecalculusofvariationsorthetheoryofgrouprepresentations)canberecast
insymplecticterms,findinginthiswaytheirultimateground.ThetheoryofHamil-
toniansystemsinitsmodernformisofcoursestilloneofthemostprominentexam-
ples.ViaHamilton-Jacobitheorythereisacloselinktothetheoryoflinearpartial
differential equations. Here, representing a differential operator on a manifold by
its symbol on the cotangent bundle and seeking solutions in terms of Lagrangian
immersions and geometrical objects living on them, one arrives at a symplectized
8See[182–184].ForahistoricaloverviewonLiegrouptheorywereferto[50]and[126].
viii Introduction
theory of first order partial differential equations.9 In a similar spirit and in close
relation,largepartsofthetheoryofsingularitieshavebeensymplectized.Another
beautifulexampleisprovidedbytheorbitmethodofKirillov,KostantandSouriau,
which constitutes one of the cornerstones of geometric quantization. For a large
class of Lie groups, this method yields a bijective correspondence between irre-
ducible unitary representations of a Lie group and transitive symplectic actions of
thisgroup.
We conclude with a few remarks on the structure and the contents of this vol-
ume.Itcontainsthreebuildingblocks,eachconsistingoffourchapters.Inthefirst
fourchapters,wepresentthecalculusonmanifolds.Thenextfourchaptersarede-
voted to the theory of Lie groups and Lie group actions and to an introduction to
linear symplectic algebra and symplectic geometry. These chapters constitute the
linkbetweentheabstractcalculusandthetheoryoffinite-dimensionalHamiltonian
systems,whichwedevelopinthefinalfourchapters.There,wehadtomakearea-
sonable selection of the topics to be presented. It is probably fair to say that our
choiceofmaterialwasmademorefromaphysicist’spointofview,10 thus,putting
emphasis on the concepts of symmetry and integrability and on Hamilton-Jacobi
theory. At the same time, this means that we had to exclude a lot of interesting
topicslike,forinstance,equivariantHamiltoniandynamicsorvariationalmethods.
Sinceeachchapterhasitsownintroduction,hereweomitadetaileddescriptionof
thecontents.
We assume that the reader is familiar with elementary algebra and calculus, as
well as with the basics of classical mechanics. Some knowledge in classical elec-
trodynamics and in thermodynamics as well as in elementary set topology will be
helpful.Thebookisself-contained,thatis,startingwiththetheoryofdifferentiable
manifolds, it guides the reader to a number of advanced topics in the theory of
Hamiltoniansystems.Atsomepoints,wetouchoncurrentresearch.Itisourstrong
beliefthatwithoutdetailedcasestudiesadeepunderstandingoftheabstractmate-
rialcanbehardlyachieved.Thus,wehaveincludedmanyworkedexamples,some
of them are taken up repeatedly. Moreover, at the end of almost every section the
readerwillfindanumberofexercises.
9ThisisduetoMaslov[197]andHörmander[141],seeChap.12ofthisvolume.Inthiscontext,
thesymplectizationofMorsetheoryplaysabasicrole,seeSects.8.9and12.4.
10Inparticular,thismeansthatwedonotgointoadvancedtopicsrelatedtosymplectictopology.
However,atsomepointswetouchonit.Forathoroughpresentationofsymplectictopologywe
refertothetextbooksofHoferandZehnder[139]andofMcDuffandSalamon[206].
Acknowledgements
It is a pleasure to thank Szymon Charzyn˙ski, Bernd Crell, Jochen Dittmann, Elis-
abeth Fischer, Christian Fleischhack, Johannes Huebschmann, Peter Jarvis, Jerzy
Kijowski, Rainer Matthes, Olaf Richter†, Matthias Schwarz, Torsten Tok, Igor P.
VolobuevandRaimarWulkenhaarforfruitfuldiscussions.Weareespeciallygrate-
ful to Christian Fleischhack, Peter Jarvis and Rainer Matthes for reading parts of
the manuscript. We also thank our students Michael Kath and Sandro Wenzel for
theirsupportinwritingthelecturenotesfromwhichthisvolumeevolved.Overthe
years,wereceivednumerousvaluableremarksonthesenotesfromstudents,notably
from Jörn Boehnke,AlexanderHertsch, André Jäschke, Adam Reichold and Axel
Schüler.
ix
Contents
1 DifferentiableManifolds . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 BasicNotionsandExamples . . . . . . . . . . . . . . . . . . . . 1
1.2 LevelSets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3 DifferentiableMappings . . . . . . . . . . . . . . . . . . . . . . . 17
1.4 TangentSpace . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.5 TangentMapping . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.6 Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.7 SubsetsAdmittingaSubmanifoldStructure . . . . . . . . . . . . . 42
1.8 Transversality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2 VectorBundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.1 TheTangentBundle . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.2 VectorBundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.3 SectionsandFrames . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.4 VectorBundleOperations . . . . . . . . . . . . . . . . . . . . . . 71
2.5 TensorBundlesandTensorFields . . . . . . . . . . . . . . . . . . 78
2.6 InducedBundles . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
2.7 SubbundlesandQuotientBundles . . . . . . . . . . . . . . . . . . 84
3 VectorFields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.1 VectorFieldsasDerivations . . . . . . . . . . . . . . . . . . . . . 94
3.2 IntegralCurvesandFlows . . . . . . . . . . . . . . . . . . . . . . 97
3.3 TheLieDerivative . . . . . . . . . . . . . . . . . . . . . . . . . . 109
3.4 Time-DependentVectorFields . . . . . . . . . . . . . . . . . . . 111
3.5 DistributionsandFoliations . . . . . . . . . . . . . . . . . . . . . 115
3.6 CriticalIntegralCurves . . . . . . . . . . . . . . . . . . . . . . . 126
3.7 ThePoincaréMapping . . . . . . . . . . . . . . . . . . . . . . . . 137
3.8 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
3.9 InvariantManifolds . . . . . . . . . . . . . . . . . . . . . . . . . 154
4 DifferentialForms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
4.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
xi