6Sub-Riemannian geometry and Lie groups. Part II.

arXiv:math/0407099v1[math.MG]7Jul2004Sub-RiemanniangeometryandLiegroups.PartII.Curvatureofmetricspaces,coadjointorbitsandassociatedrepresentationsMariusBuligaIMBBˆatimentMA´EcolePolytechniqueF´ed´eraledeLausanneCH1015Lausanne,SwitzerlandMarius.Buliga@epfl.chandInstituteofMathematics,RomanianAcademyP.O.BOX1-764,RO70700Bucure¸sti,[e-mail]:05.07.20041Introduction22Distancesandmetricprofiles42.1Distancesbetweenmetricspaces......................42.2Metricprofiles................................53Sub-Riemannianmanifolds63.1Normalframesandprivilegedcharts....................73.2Thenilpotentizationatapoint.......................104Deformationsofsub-Riemannianmanifolds125MeaningofMitchelltheorem1136Tangentbundlesanddilatationstructures147Curvatureofanicemetricprofile178Homogeneousspacesandmotivationforcurvature188.1Riemanniansurfaces.............................218.2AboutCartanmethodofequivalenceforsub-Riemannianmanifolds..228.3Homogeneouscontact3manifolds.....................239Homogeneousensemblesanddeformations2610Whatcurvatureis2911Curvatureandcoadjointorbits3012Appendix:Uniformandconicalgroups351IntroductionThispaperisthethirdinaseries[4],[5],dedicatedtothefundamentalsofsub-RiemanniangeometryanditsimplicationsinLiegroupstheory.Wealsobringtotheattentionofthereaderthepaper[7]forthemoreanalyticalaspectsofthetheory.Thepointofviewofthesepapersisthatwhatwearedealingwitharemanifestationsoftheemergingnon-Euclideananalysis.ThisbecomevisibleespeciallyinthemetricstudyofLiegroupsendowedwithleftinvariantgeneratingdistributions.Metricprofilesandtheircurvaturefirstappearedin[6].Theygivenewinsightsintothebehaviourofmetricspaces.Thesubjectisincloselinkwithsub-RiemannianLiegroupsbecausecurvaturescouldbeclassifiedbycomparisonwith(metricprofilesof)homogeneousspaces.ThehomogeneousspacesweareinterestedincanbeseenasfactorspacesofLiegroupswithleftinvariantdistributions.Thecurvatureofametricspaceinapointis,bydefinition,therectifiabilityclassofthemetricprofileassociatedtothepoint.Therearetwoproblemsherethatwearetryingtosolve.Thefirstproblemcomesfromthefactthattoametricspacewithsomegeometricstructurewecanassociateseveralmetricprofiles.Themostinterestingaretheprofileasametricspaceandthedilatationprofile,associatedwiththedilatationstructureofthespace.Thedilatationstructureisthebasicobjectinthestudyofdifferentialpropertiesofmetricspacesofacertaintype.(ForexampleanyRiemannianorsub-Riemannianmanifold,withorwithoutconicalsingularities,hasadilatationstructure.Buttherearemetricspaceswhicharenotadmittingmetrictangentspaces,buttheyadmitdilatationstructures).Thisstructuretellsuswhatisthegoodnotionofanalysisonthatspace.Thereisaninfiniteclassofdifferentsuchanalysisandtheclassicalone,whichwecallEuclidean,isonlyoneofthem.Toeachmetricprofileofthespaceweassociateacurvature.Weshallhavethereforeametriccurvatureandadilatationcurvature.Themetriccurvatureismoredifficulttodescribeandinfactthedilatationcurvaturecontainsmorecomplexinformations.Thefirstproblemrelatedtocurvatureis:whatcurvaturewemeasure?Thesecondproblemofcurvatureisrelatedtothefactthatthedefinitionofcurvatureasarectifiabilityclassistooabstracttoworkwith,unlesswehavea”good”setofrepresentativesfortherectifiabilityclasses.Supposethatwehavemadesuchachoiceofthe”good”classofrepresentatives.Thenweshallsaythatthecurvatureofaspaceinapointistherepresentantoftherectifiabilityclass.Thesecondproblemofcurvatureis:whatisa”good”classofrepresentatives?Thisisobviouslyasubjectivematter;anywayinthispaperthissubjectivityisrevealedand,oncerevealed,theproposedchoicesbecomelesssubjective.Thegoalofthispaperistoshowthatcurvatureinthesenseofrectifiabilityclassofthedilatationprofilecanbeclassifiedusingcoadjointorbitsrepresentations.Weestablishhereabridgebetween:-curvaturenotioninspaceswhichadmitmetrictangentspacesatanypoint,and-self-adjointrepresentationsofalgebrasnaturallyassociatedwiththestructureofthetangentspace.Inouropinionthisisthesecondlinkbetweensub-Riemanniangeometryandquan-tummechanics.ThefirstlinkcanbeuncoveredfromBuliga[5]section5”CaseoftheHeisenberggroup”,asexplainedinthesection11ofthepresentpaper.Anappendixconcerninguniformandconicalgroupsisaddedtothepaper.MoreinformationaboutthesubjectcanbefoundinBuliga[5],sections3and4.2DISTANCESANDMETRICPROFILES42DistancesandmetricprofilesThereferencesforthefirstsubsectionareGromov[12],chapter3,Gromov[10],andBurago&al.[8]section7.4.Thereareseveraldefinitionsofdistancesbetweenmetricspaces.TheveryfertileideaofintroducingsuchdistancesbelongstoGromov.2.1DistancesbetweenmetricspacesInordertointroducetheHausdorffdistancebetweenmetricspaces,recalltheHausdorffdistancebetweensubsetsofametricspace.Definition2.1ForanysetA⊂Xofametricspaceandanyε0settheεneigh-bourhoodofAtobeAε=[x∈AB(x,ε)TheHausdorffdistancebetweenA,B⊂XisdefinedasdXH(A,B)=inf{ε0:A⊂Bε,B⊂Aε}ByconsideringallisometricembeddingsoftwometricspacesX,YintoanarbitrarymetricspaceZweobtaintheHausdorffdistancebetweenX,Y(Gromov[12]definition3.4).Definition2.2TheHausdorffdistancedH(X,Y)betweenmetricspacesXYistheinfimumofthenumbersdZH(f(X),g(Y))fo