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1、MarcelBergerAPanoramicViewofRiemannianGeometry6thSeptember2002SpringerBerlinHeidelbergNewYorkBarcelonaHongKongLondonMilanParisTokyoPrefaceVIIIPrefaceRiemanniangeometryhasbecomeanimportantandvastsubject.Itdeservesanencyclopedia,ratherthanamodestlengthbook.Itisthereforeimpossibletopre
2、sentRiemanniangeometryinabookinthestandardfashionofmath-ematics,withcompletedefinitions,proofs,andsoon.Thiscontrastssharplywiththesituationin1943,whenPreissmann’sdissertation1943[1041]pre-sentedalltheglobalresultsofRiemanniangeometry(butforthetheoryofsymmetricspaces)includingnewones,
3、withproofs,inonlyfortypages.Moreover,evenattherootofthesubject,theideaofaRiemannianmani-foldissubtle,appealingtounnaturalconcepts.Consequently,allrecentbooksonRiemanniangeometry,howevergoodtheymaybe,canonlypresenttwoorthreetopics,havingtospendquiteafewpagesonthefoundations.Sinceoura
4、imistointroducethereadertomostofthelivingtopicsofthefield,wehavehadtofollowtheonlypossiblepath:topresenttheresultswithoutproofs.Ouraimistwofold,asannouncedbyoursubtitle.ThefirstistointroducethevariousconceptsandtoolsofRiemanniangeometryinthemostnaturalway;ormore,todemonstratethatoneis
5、practicallyforcedtodealwithabstractRiemannianmanifoldsinahostofintuitivegeometricalquestions.Thisexplainstheword“introduction”andthefactthatalongfirstchapterwilldealwithproblemsintheEuclideanplane.Secondly,onceequippedwiththeconceptofRiemannianmanifold,wewillpresentapanoramaofcurrent
6、dayRiemanniangeometry.Apanoramaisneverafull360degrees,sowewillnotpretendtobecomplete,buthopethatourpanoramawillbelargeenoughtoshowthereaderasubstantialpartoftoday’sRiemanniangeometry.Andinapanorama,youseethepeaks,butyoudonotclimbthem.Thisisawaytosay,bycaricature,thatwewillnotproveth
7、estatementswequote.But,inapanorama,sometimesyoucanstillseethepathtoasummit.Herethismeansthatinmanycaseswewillexplainthemainideasorthemainingredientsfortheproof.Wehopethatthiswayofwritingwillleavemanyreaderswantingtoclimbsomepeak.Wewillgivealltheneededreferencestotheliteratureasthein
8、troductionandthepan
