On the flag curvature of Finsler metrics of scalar

arXiv:math/0303144v1[math.DG]12Mar2003OntheFlagCurvatureofFinslerMetricsofScalarCurvature∗XinyueChen†,XiaohuanMo‡andZhongminShenMarch15,2003AbstractTheflagcurvatureofaFinslermetriciscalledaRiemannianquantitybecauseitisanextensionofsectionalcurvatureinRiemanniangeometry.InFinslergeometry,thereareseveralnon-Riemannianquantitiessuchasthe(mean)Cartantorsion,the(mean)LandsbergcurvatureandtheS-curvature,whichallvanishforRiemannianmetrics.Itisimportanttounderstandthegeometricmeaningsofthesequantities.Inthispaper,westudyFinslermetricsofscalarcurvature(i.e.,theflagcurvatureisascalarfunctionontheslittangentbundle)andpartiallydeterminetheflagcurvaturewhencertainnon-Riemannianquantitiesareisotropic.Usingtheobtainedformulafortheflagcurvature,weclassifylocallyprojectivelyflatRandersmetricswithisotropicS-curvature.1IntroductionFinslermetricsarisenaturallyfrommanyareasofmathematicsaswellasnaturalscience.Forexample,thenavigationprobleminaRiemannianspacegivesrisetoalotsofinterestingFinslermetricswithspecialgeometricproperties[28][29][30].InFinslergeometry,westudynotonlytheshapeofaspace,butalsothe“color”ofthespaceonaninfinitesimalscale.TheRiemannianquantity(suchastheflagcurvature)describestheshapeofaspace,whilenon-Riemannianquantitiesdescribesthe“color”ofthespace.ForaFinslermanifold(M,F),theflagcurvatureK=K(P,y)isafunctionoftangentplanesP=span{y,v}⊂TxManddirectionsy∈P\{0}.Thisquantitytellsushowcurvedthespaceisatapoint.IfFisRiemannian,K=K(P)isindependentofy∈P\{0},KbeingcalledthesectionalcurvatureinRiemanniangeometry.AFinslermetricFissaidtobeofscalarcurvatureiftheflagcurvatureK=K(x,y)isascalarfunctionontheslittangentbundle∗2000MathematicsSubjectClassification:Primary53B40,53C60†supportedbytheNationalNaturalScienceFoundationofChina(10171117)‡supportedbytheNationalNaturalScienceFoundationofChina(10171002)1TM\{0}.Clearly,aRiemannianmetricisofscalarcurvatureifandonlyifK=K(x)isascalarfunctiononM(whichisaconstantindimensionn2bytheSchurlemma).Therearelotsofnon-RiemannianFinslermetricsofscalarcurvature.OneoftheimportantproblemsinFinslergeometryistostudyandcharacterizeFinslermetricsofscalarcurvature.Thisproblemhasnotbeensolvedyet,evenforFinslermetricsofconstantflagcurvature.AccordingtoE.Cartan’slocalclassificationtheorem,anyRiemannianmetricαofconstantsectionalcurvatureμislocallyisometrictothefollowingstandardmetricαμontheunitballBn⊂RnorthewholeRnforμ=−1,0,+1:α−1(x,y)=p|y|2−(|x|2|y|2−hx,yi2)1−|x|2,y∈TxBn∼=Rn,(1)α0(x,y)=|y|,y∈TxRn∼=Rn,(2)α+1(x,y)=p|y|2+(|x|2|y|2−hx,yi2)1+|x|2y∈TxRn∼=Rn.(3)Thesimplestnon-RiemannianFinslermetricsarethoseintheformF=α+β,whereαisaRiemannianmetricandβisa1-form.TheyarecalledRandersmetrics.Bao-RoblesprovethatifaRandersmetricF=α+βhasisotropicflagcurvatureK=K(x),thenthereisaconstantcsuchthatthecovariantderivativesofβwithrespecttoαsatisfyasystemofPDEs[4](i.e.,equation(35)belowwithc(x)=c).Recently,Bao-Robles-ShenhaveclassifiedRandersmetricsofconstantcurvatureviathenavigationprobleminRieman-nianmanifolds[5].AFinslermetricissaidtobelocallyprojectivelyflatifatanypointthereisalocalcoordinatesysteminwhichthegeodesicsarestraightlinesaspointsets.WhyareweinterestedinthesetypeofFinslermetrics?Riemannianmetricsofconstantcurvaturearelocallyprojectivelyflat.TheconverseistruetooaccordingtoBeltrami’stheorem.ProjectivelyflatFinslermetricsonaconvexdomaininRnareregularsolutionstoHilbert’sFourthProblem[18].ItisknownthateverylocallyprojectivelyflatFinslermetricisofscalarcurvature.LocallyprojectivelyflatFinslermetricswithconstantflagcurvaturehavebeensolvedatsatisfactorylevel[7][8][9]-[11],[15]-[17],[26],[27].InFinslergeometry,thereareseveralimportantnon-Riemannianquantities:thedistortionτ,themeanCartantorsionI,theS-curvatureSandthemeanLandsbergcurvatureJ,etc.TheyallvanishforRiemannianmetrics,hencetheyaresaidtobenon-Riemannian.SeeSection2formoredetailsabouttheirdefinitionsandgeometricmeanings.AllknownRandersmetricsF=α+βofscalarcurvature(indimensionn2)satisfyS=(n+1)c(x)ForJ+c(x)FI=0,wherec(x)isascalarcurvature(seeTheorem1.3belowforprojectivelyflatexamples).InordertoclassifyFinslermetricsofscalarcurvature,wefirstinvestigatethosewithisotropicS-curvature.2Theorem1.1Let(M,F)beann-dimensionalFinslermanifoldofscalarcur-vaturewithflagcurvatureK(x,y).SupposethattheS-curvatureisisotropic,S=(n+1)c(x)F(x,y),(4)wherec(x)isascalarfunctiononM.Thenthereisascalarfunctionσ(x)onMsuchthatK=3cxm(x)ymF(x,y)+σ(x).(5)Inparticular,c(x)=cisaconstantifandonlyifK=K(x)isascalarfunctiononM.In(5)andthereafter,thesubscriptxmincxmindicatespartialdifferentiationwithrespecttoxm.InTheorem1.1,wepartiallydeterminetheflagcurvaturewhentheS-curvatureisisotropic.Thisisageneralizationofatheoremin[22]wherethesecondauthorshowsthattheflagcurvatureisisotropic,K=K(x)if(4)holdsforc(x)=constant.Inthiscase,K=constantwhenn≥3bytheSchurtheorem[3].Theorem1.2Let(M,F)beann-dimensionalFinslermanifoldofscalarcur-vature.SupposethatJ/Iisisotropic,J+c(x)FI=0,(6)wherec=c(x)isaC∞scalarfunctiononM.ThentheflagcurvatureK=K(x,y)andthedistortionτ=τ(x,y)satisfyn+13Kyk+K+c(x)2−cxm(x)ymF(x,y)τyk=0.(7)(a)Ifc(x)=cisaconstant,thenther