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1、DiRerentia1GeometryandSymmetricSpacesPUREANDAPPLIEDMATHEMATICSASeriesofMonographsandTextbooksEditedbyPAULA.SMITHandSAMUELEILENBERGColumbiaUniversity,NewYorkI:ARNOLDSOMMERFELD.PartialDifferentialEquationsinPhysics.1949(LecturesonTheoreticalPhysics,Volu
2、meVI)11:REINHOLDBAER.LinearAlgebraandProjectiveGeometry.1952111:HERBERTBUSEMANNANDPAULJ.KELLY.ProjectiveGeometryandProjectiveMetrics.1953IV:STEFANBERGMANANDM.SCHIFFER.KernelFunctionsandEllipticDifferentialEquationsinMathematicalPhysics.1953V:RALPHPHIL
3、IPBOAS,JR.EntireFunctions.1954VI:HERBERTBUSEMANN.TheGeometryofGeodesics.1955VII:CLAUDECHEVALLEY.FundamentalConceptsofAlgebra.1956VIII:SZE-TSENHu.HomotopyTheory.1959IX:A.OSTROWSICI.SolutionofEquationsandSystemsofEquations.1960X:J.DIEUDONN~.Foundationso
4、fModernAnalysis.1960XI:S.I.GOLDBERG.CurvatureandHomology.1962XII:SIGURDURHELGASON.DifferentialGeometryandSymmetricSpaces.1962InpreparationXIII:T.H.HILDEBRANDT.IntroductiontotheTheoryofIntegration.DifferentialGeometrySymmetricSpaces5igurdurHelgasonDepa
5、rtmentofMathematics,MassachusettsInstituteofTechnology,Cambridge,Massachusetts1962ACADEMICPRESSNEWYORKANDLONDONCOPYRIGHT01962,BYACADEMICPRESSINC.ALLRIGHTSRESERVEDNOPARTOFTHISBOOKMAYBEREPRODUCEDINANYFORMBYPHOTOSTAT,MICROFILMORANYOTHERMEANS,WITHOUTWRITT
6、ENPERMISSIONFROMTHEPUBLISHERSACADEMICPRESSINC.111FIFTHAVENUENEWYORK3,N.Y.UnitedKingdomEditionPublishedbyACADEMICPRESSINC.(LONDON)LTD.BerkeleySquareHouse,London,IN.1LibraryofCongressCatalogCardNztniber62-13107PRINTEDINTHEUNITEDSTATESOFAMERICAToArtieThi
7、sPageIntentionallyLeftBlankPREFACEAccordingtoitsoriginaldefinition,asymmetricspaceisaRiemannianmanifoldwhosecurvaturetensorisinvariantunderallparalleltrans-lations.ThetheoryofsymmetricspaceswasinitiatedbyE.Cartanin1926andwasvigorouslydevelopedbyhimint
8、helate1920’s.Bytheirdefinition,symmetricspacesformaspecialtopicinRiemanniangeometry;theirtheory,however,hasmergedwiththetheoryofsemisimpleLiegroups.Thiscircumstanceisthesourceofverydetailedandextensiveinformationaboutthesespaces.Theycantherefo
