Curvature of sub-Riemannian spaces

arXiv:math/0311482v1[math.MG]26Nov2003Curvatureofsub-RiemannianspacesMariusBuligaInstitutBernoulliBˆatimentMA´EcolePolytechniqueF´ed´eraledeLausanneCH1015Lausanne,SwitzerlandMarius.Buliga@epfl.chandInstituteofMathematics,RomanianAcademyP.O.BOX1-764,RO70700Bucure¸sti,[e-mail]:26.11.200311INTRODUCTION21IntroductionToanymetricspacesthereisanassociatedmetricprofile.Therectifiabilityofthemetricprofilegivesagoodnotionofcurvatureofasub-Riemannianspace.Weshallsaythatacurvatureclassistherectifiabilityclassofthemetricprofile.Weclassifythenthecurvaturesbylookingtohomogeneousmetricspaces.Theclassificationprob-lemissolvedforcontact3manifolds,wherewerediscovera3dimensionalfamilyofhomogeneouscontactmanifolds,withadistinguished2dimensionalfamilyofcontactmanifoldswhichdon’thaveanaturalgroupstructure.Theclassificationof3dimen-sionalhomogeneouscontactmanifoldshasbeendonebyHughen[8].IhavediscoveredmetricprofilesinvariousproofofMitchelltheorem1.Alsothisisexplainedinthepaper.Inmyopinion,theuseofthenotionofmetricprofileclarifiesthequestion:whyseveralproofsforsameresult(Mitchelltheorem1)andmoreover,anyofthemequallylongandcomplex?Ithastobementionedthatcontrarytootherattemptstodefinethecurvatureofasub-Riemannianmanifold,hereispresentedanalmostpuremetricalconstruction,notusingdifferentialgeometry,whichisnotoriouslymisleadingwhenusedinasub-Riemannianframe.Onceoneknowswhattolookfor,thendifferentialgeometry(read”Euclideananalyticdifferentialgeometry”)recoversitswellknownstrength,though.Thestructureofthepaperisdescribedfurther.Insections2–5isgivenashortpre-sentationofsub-Riemannianmanifolds,Carnotgroups,PansuderivativeandGromov-Hausdorffdistance.Fortheexpertreaderthesesectionsserveonlytofixnotationsneededlater.Section5isaboutdeformationsofsub-Riemannianmanifold,seenascurvesinthespaceCMSofisometryclasssesofcompactmetricspaces,withtheGromov-Hausdorffdistance.Insection6canbefoundadiscussionofvariousproofsofMitchell[11]theorem1.Thissectionjustifiesthenotionofmetricprofile,whichisthesubjectofsection7.Inthesamesectionisgiventheenotionofcurvatureintermsofrectifiabilityclassesofmetricprofiles.Inordertoclassifythecurvatureshomogeneousspacesareused.Section8isdedi-catedtothissubject.Asanapplication,insection9arestudiedthehomogeneouscontact3manifolds,Finally,insection10theproblemofclassificationissolvedforalargeclassofcontact3manifolds.2Regularsub-RiemannianmanifoldsClassicalreferencestothissubjectareBella¨ıche[1]andGromov[6].Theinterestedreaderisadvisedtolookalsotothereferencesofthesepapers.LetMbeaconnectedmanifold.Adistribution(orhorizontalbundle)isasubbundleDofM.Toanypointx∈MthereisassociatedthevectorspaceDx⊂TxM.GiventhedistributionD,apointx∈Mandasufficientlysmallopenneighbour-hoodx∈U⊂M,onecandefineonUafiltrationofbundlesasfollows.Definefirst2REGULARSUB-RIEMANNIANMANIFOLDS3theclassofhorizontalvectorfieldsonU:X1(U,D)={X∈Γ∞(TU):∀y∈U,X(u)∈Dy}Next,defineinductivelyforallpositiveintegersk:Xk+1(U,D)=Xk(U,D)∪[X1(U,D),Xk(U,D)]Here[·,·]denotesvectorfieldsbracket.WeobtainthereforeafiltrationXk(U,D)⊂Xk+1(U,D).Evaluatenowthisfiltrationatx:Vk(x,U,D)=nX(x):X∈Xk(U,D)oTherearem(x),positiveinteger,andsmallenoughUsuchthatVk(x,U,D)=Vk(x,D)forallk≥mandDxV1(x,D)⊂V2(x,D)⊂...⊂Vm(x)(x,D)Weequallyhaveν1(x)=dimV1(x,D)ν2(x)=dimV2(x,D)...n=dimMGenerallym(x),νk(x)mayvaryfromapointtoanother.Thenumberm(x)iscalledthestepofthedistributionatx.Definition2.1ThedistributionDisregularifm(x),νk(x)areconstantontheman-ifoldM.Thedistributioniscompletelynon-integrableifforanyx∈MwehaveVm(x)=TxM.Definition2.2Asub-Riemannian(SR)manifoldisatriple(M,H,g),whereMisaconnectedmanifold,Hisacompletelynon-integrabledistributiononM,andgisametric(Euclideaninner-product)onthehorizontalbundleH.TheCarnot-Carath´eodorydistanceassociatedtothesub-Riemannianmanifoldisthedistanceinducedbythelengthlofhorizontalcurves:d(x,y)=inf{l(c):c:[a,b]→M,c(a)=x,c(b)=y}TheChowtheoremensurestheexistenceofahorizontalpathlinkinganytwosuf-ficientlyclosedpoints,thereforetheCCdistanceitatleastlocallyfinite.Weshallworkfurtheronlywithregularsub-Riemannianmanifolds,ifnototherwisestated.Bella¨ıcheintroducedtheconceptofprivilegedchartaroundapointp∈M.Let(x1,...,xn)7→φ(x1,...,xn)∈MbeachartofMaroundp(i.e.phascoordinates(0,....,0)).DenotebyX1,...,Xntheframeofvectorfieldsassociatedtothecoordinatechart.Thechartiscalledadapted(ortheframeiscalledadapted)ifthefollowinghappens:X1,...,Xν1formsabasisofV1,Xν1+1,...,Xν2formabasisofV2,andsoon.SupposethattheframeX1,...,Xnisadapted.ThedegreeofXiisthenkifXi∈Vk\Vk−1.3CARNOTGROUPS4Definition2.3Achart(oraframe)isprivilegedifmoreoverthefollowinghappens:foranyi=1,...,nthefunctiont7→d(p,φ(...,t,...))(withtonthepositioni)isexactlyoforderdegXiatt=0.Privilegedcharts(frames)alwaysexist,asprovedbyBella¨ıche[1]Theorem4.15.Aprivilegedframetransformsthefiltrationintoadirectsum.DefineVi=span{Xk:degXk=i}ThenthetangentspacedecomposesasadirectsumofvectorspacesVi.Moreover,eachspaceVidecomposesinadirectsumofspacesVkwithk≤i.Theintrinsicdilatationsassociatedtoaprivilegedframearedefined,inthechartφ,foranyε0(sufficientlysmallifnecessar