Reconstructing the geometric structure of a Rieman

arXiv:0801.4127v1[math.DG]27Jan2008ReconstructingthegeometricstructureofaRiemanniansymmetricspacefromitsSatakediagramSebastianKlein1January24,2008Abstract.ThelocalgeometryofaRiemanniansymmetricspaceisdescribedcompletelybytheRiemannianmetricandtheRiemanniancurvaturetensorofthespace.InthepresentarticleIdescribehowtocomputethesetensorsforanyRiemanniansymmetricspacefromitsSatakediagram,inawaythatissuitedfortheusewithcomputeralgebrasystems;anexampleimplementationforMapleVersion10canbefoundon(3)/SO(3)areclassified.Author’saddress.Sebastian[e-mail]:Satakediagram,structureconstants,Chevalleyconstants,curvaturetensor,Rie-manniansymmetricspaceMSclassificationnumbers:53C35(Primary);53B20,17B20,17-081IntroductionItiswell-knownthatthebehaviorofanyRiemannianmanifoldMisinfluencedstronglybyitsRiemanniancurvaturetensor.Togivejusttwoexamples,the“spreading”ofthegeodesics,asmeasuredbytheJacobifields,andtheexistenceofatotallygeodesicsubmanifoldtangentialtoagivensubspaceofatangentspaceareexpressedintermsofthecurvaturetensorfieldonthemanifoldM.Especiallyfor(locally)Riemanniansymmetricspaces,thecontrolexertedonthelocalgeo-metryofthemanifoldbytheRiemanniancurvature,togetherwiththeRiemannianmetric,istotal:IfMandNaretwosuchspaces,andthereexistsalinearisometryTpM→TqNwhich1ThisworkwassupportedbyafellowshipwithinthePostdoc-ProgrammeoftheGermanAcademicExchangeService(DAAD).11IntroductiontransportsthecurvaturetensorofMatpintothecurvaturetensorofNatq,thenMandNarealreadylocallyisometrictoeachother;notethatitsufficestoconsiderthecurvatureinasinglepointofMandNbecausethecurvaturetensorfieldisparallelinthissituation.ThisshowsthatforaRiemanniansymmetricspaceM,thelocalgeometryofMisdescribedcompletelybytwotensorsonasingletangentspaceTpM:theinnerproductgivenbytheRie-mannianmetricatp,andthecurvaturetensoratp.Viewedinthisway,thestudyofthelocalgeometryofaRiemanniansymmetricspacereducestoapurelyalgebraicproblem,namelytothestudyofthesetwotensorsonthetangentspaceTpM.Thuswewillcallthesetwotensorsthe“fundamentalgeometrictensors”ofM.OneveryimportantexampleforthecontrolofthegeometryofaRiemanniansymmetricspaceMbyitscurvaturetensorRisthefollowingresult,whichpermitstheclassificationofthetotallygeodesicsubmanifoldsofM:AlinearsubspaceU⊂TpMisthetangentspaceofatotallygeodesicsubmanifoldofMifandonlyifUisaLietriplesystem,i.e.ifR(u,v)w∈Uholdsforallu,v,w∈U.Inviewoftheabove,itisverydesirabletohaverepresentationsofthefundamentalgeometrictensors,especiallythecurvaturetensor,availableforstudyforeveryRiemanniansymmetricspaceM=G/K.Thewell-knownformulaR(u,v)w=−[[u,v],w]relatingthecurvaturetensorRofMtotheLiebracketoftheLiealgebragofthetransvectiongroupGofMletsonecalculateRrelativelyeasilyifGisaclassicalgroup(thengisamatrixLiealgebra,withtheLiebracketbeingsimplythecommutatorofmatrices),butnotsoeasilyifGisoneoftheexceptionalLiegroups,becausethentheexplicitdescriptionofgasamatrixalgebraistoounwieldytobeusefulgenerally.Therefore,inthepresentpaper,IwilldescribeanotherrepresentationofthefundamentalgeometrictensorsofanyRiemanniansymmetricspaceofcompacttype,basedontherootspacedecompositionoftheLiealgebraofitstransvectiongroup.Thisrepresentationisespeciallysuitedfortheusewithacomputeralgebrasystem.IhaveimplementedthealgorithmsandequationsgivenhereasaMaplepackage,whichcanbefoundat(basedonsimilarmethodsasmyclassificationinthe2-Grassmannians,see[K1]and[K2]).Asinformationaboutthesymmetricspaceconcerned,wewillrequireonlytheSatakediagramofthatspace,i.e.theDynkindiagramoftheLiealgebraofthetransvectiongroup,“annotated”withfurtherinformationdescribingthesymmetricstructureofthespace,seeforexample[Lo],SectionVII.3.3,p.132ff..TheSatakediagramsarewell-knownandtabulatedintheliterature(forexample,in[Lo],p.147f.)foreveryirreducibleRiemanniansymmetricspace.Itisawell-knownfactthattheSatakediagramalreadydeterminesthelocalstructureoftheRiemanniansymmetricspace;however,itturnsoutthatfortheactualreconstructionofthefundamentalgeometrictensorsinasufficientlyexplicitway,somenewworkneedstobedone.Ourconsiderationisbasedonthefollowingwell-knownconstruction:LetusconsideraRie-manniansymmetricspaceM=G/Kofcompacttype.ThenthesymmetricstructureofMinducesaninvolutiveautomorphismσontheLiealgebragofthetransvectiongroupGofM,andσgivesrisetothedecompositiong=k⊕m,wherek=Eig(σ,1)istheLiealgebraoftheisotropygroupKandm=Eig(σ,−1)isalinearsubspaceofgwhichiscanonicallyisomorphictothetangentspaceToMatthe“originpoint”o:=eK∈G/K=M,andifweidentifyToMwithminthatway,thenoneachirreduciblefactorofM,theRiemannianmetricofMisa21IntroductionconstantmultipleoftheKillingformofg,andthecurvaturetensorRofMatoisgivenbytheformulaR(u,v)w=−[[u,v],w]foru,v,w∈m.ToreconstructthecurvaturetensorandtheinnerproductonToMf