foundations differential geometry - michor

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1、FOUNDATIONSOFDIFFERENTIALGEOMETRYPeterW.MichorMailingaddress:PeterW.Michor,InstitutfurMathematikderUniversitatWien,Strudlhofgasse4,A-1090Wien,Austria.E-mailmichor@awirap.bitnetThesenotesarefromalecturecourseDierentialgeometrieundLieGruppenwhichhasbeenheldattheUniversit

2、yofViennaduringtheacademicyear1990/91,againin1994/95,andinWS1997.Itisnotyetcompleteandwillbeenlargedduringtheyear.InthislecturecourseIgivecompletedenitionsofmanifoldsinthebeginning,but(besidespheres)examplesaretreatedextensivelyonlylaterwhenthetheoryisdevelopedenough.Ia

3、dviseeverynovicetotheeldtoreadtheexcellentlecturenotesTypesetbyAS-TXMEiiTABLEOFCONTENTS1.DierentiableManifolds.....................12.SubmersionsandImmersions..................133.VectorFieldsandFlows....................184.LieGroupsI.........................395.LieGro

4、upsII.LieSubgroupsandHomogeneousSpaces.......556.VectorBundles........................647.DierentialForms.......................768.IntegrationonManifolds....................849.DeRhamcohomology.....................9110.CohomologywithcompactsupportsandPoincareduality..

5、....................10211.DeRhamcohomologyofcompactmanifolds...........11712.LiegroupsIII.AnalysisonLiegroups..............12413.DerivationsontheAlgebraofDierentialFormsandtheFrolicher-NijenhuisBracket...............13714.FiberBundlesandConnections.................14615

6、.PrincipalFiberBundlesandG-Bundles.............15716.PrincipalandInducedConnections...............17517.Characteristicclasses.....................19618.Jets.............................212References............................219ListofSymbols.........................222I

7、ndex..............................224DraftfromNovember17,1997PeterW.Michor,11.DierentiableManifolds1.1.Manifolds.AtopologicalmanifoldisaseparablemetrizablespaceMnwhichislocallyhomeomorphictoR.Soforanyx2Mthereissomehomeo-nmorphismu:U!u(U)R,whereUisanopenneighborhoodofxi

8、nMandnu(U)isanopensubsetinR.Thepair(Uu)iscalledachartonM.Fromalgebraictopologyitfollowsthatthenumberni