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buliga m. sub-riemannian geometry and lie groups, part 1 (seminar notes, dma-epfl, 2001, math.mg_0210189, 2002)(80s)

buliga m. sub-riemannian geometry and lie groups, part 1 (seminar notes, dma-epfl, 2001, math.mg_0210189, 2002)(80s)

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时间：2018-07-26

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1、Sub-RiemanniangeometryandLiegroupsPartISeminarNotes,DMA-EPFL,2001M.BuligaInstitutBernoulliB^atimentMAEcolePolytechniqueFederaledeLausanneCH1015Lausanne,SwitzerlandMarius.Buliga@ep .chandInstituteofMathematics,RomanianAcademyP.O.BOX1-764,RO70700Bucuresti,Roman

2、iaMarius.Buliga@imar.roThisversion:October31,2002Keywords:sub-Riemanniangeometry,symplecticgeometry,CarnotgroupsarXiv:math.MG/0210189v331Oct200212IntroductionM.Gromov[13],pages85{86:"3.15.Proposition:Let(V;g)beaRiemannianmanifoldwithgcontinuous.Foreachv2Vthespace

3、s(V;v)Lipschitzconvergeas!1tothetangentspace(TvV;0)withitsEuclideanmetricgv.Proof+:StartwithaC1map(Rn;0)!(V;v)whosedi erentialisisometricat0.The-scalingsofthisprovidealmostisometriesbetweenlargeballsinRnandthoseinVfor!1.Remark:Infactwecande neRiemannianmanif

4、oldsaslocallycompactpathmetricspacesthatsatisfytheconclusionofProposition3.15."Ifso,Gromov'sremarkshouldapplytoanysub-Riemannianmanifold.Whythenisthesub-Riemanniancasesodi erentfromtheRiemannianone?Hereisalistoflegitimatequestions:Howcanonede nethemanifoldstructu

5、re?Whoarethetangentandcotangentbundles?Whatistheintrinsicdi erentialcalculus?WhyarethereabnormalgeodesicsiftheHamiltonianformalismonthecotangentbundlewerecomplete?IfthemanifoldisacompactLiegroupdoesthetangentbundlecarryanaturalgroupstructure?Whataredi erentialfor

6、ms,deRhamcochain,andthevariationalcomplex?Considerthegroupofsmoothvolumepreservingtransformations.Whydoesthisgrouphavemoreinvariantsthanthevolumeandwhatistheinterpretationoftheseinvariants?Thepurposeofthisworkingseminarwastoexploreasmanyaspossibleopenquestionsfro

7、mthelistabove.SpecialattentionhasbeenpayedtothecaseofaLiegroupwithaleftinvariantdistribution.Theseminar,organisedbytheauthorandTudorRatiuattheMathematicsDe-partment,EPFL,startedinNovember2001.Thepaperisbynomeansselfcontained.Foranyunprovedresultitisindicatedthep

8、lacewhereacompleteproofcanbefound.Thechoiceoftheproofsisratherpsychological:someofthemhaveageometricalormixedgeometric-analyticalmean-ing(liketheproofofHopf-Ri

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