Invariance Theory, the Heat Equation and the Atiyah-Singer Index Theorem英文文献资料

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1、INVARIANCETHEORY,THEHEATEQUATION,ANDTHEATIYAH-SINGERINDEXTHEOREMbyPeterB.GilkeyElectronicreprint,copyright1996,PeterB.GilkeyBookoriginallypublishedonpaperbyPublishorPerishInc.,USA,1984LibraryofCongressCatalogCardNumber84-061166ISBN0-914098-20-9INTRODUCTIONThisbooktre

2、atstheAtiyah-Singerindextheoremusingheatequationmethods.Theheatequationgivesalocalformulafortheindexofanyellipticcomplex.Weuseinvariancetheorytoidentifytheintegrandoftheindextheoremforthefourclassicalellipticcomplexeswiththeinvari-antsoftheheatequation.Sincethetwiste

3、dsignaturecomplexprovidesasucientlyrichfamilyofexamples,thisapproachyieldsaproofoftheAtiyah-Singertheoremincompletegenerality.WealsouseheatequationmethodstodiscussLefschetzxedpointformulas,theGauss-Bonnetthe-oremforamanifoldwithsmoothboundary,andthetwistedetainvari

4、ant.Weshallnotincludeadiscussionofthesignaturetheoremformanifoldswithboundary.Therstchapterreviewsresultsfromanalysis.Sections1.1through1.7representstandardellipticmaterial.Sections1.8through1.10containthematerialnecessarytodiscussLefschetzxedpointformulasandothert

5、op-ics.Invariancetheoryanddierentialgeometryprovidethenecessarylinkbe-tweentheanalyticformulationoftheindextheoremgivenbyheatequationmethodsandthetopologicalformulationoftheindextheoremcontainedintheAtiyah-Singertheorem.Sections2.1through2.3areareviewofchar-acterist

6、icclassesfromthepointofviewofdierentialforms.Section2.4givesaninvariant-theoreticcharacterizationoftheEulerformwhichisusedtogiveaheatequationproofoftheGauss-Bonnettheorem.Sections2.5and2.6discussthePontrjaginformsofthetangentbundleandtheChernformsofthecoecientbundl

7、eusinginvariancetheory.ThethirdchaptercombinestheresultsofthersttwochapterstoprovetheAtiyah-Singertheoremforthefourclassicalellipticcomplexes.WerstpresentaheatequationproofoftheHirzebruchsignaturetheorem.Thetwistedspincomplexprovidesauniedwayofdiscussingthesignatu

8、re,Dolbeault,anddeRhamcomplexes.Insections3.2{3.4,wediscussthehalf-spinrepresentations,thespincomplex,andderiveaformulaforthe^Agenu