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ID:38817132
大小:2.32 MB
页数:580页
时间:2019-06-19
《Lectures on the geometry of manifolds英文学习资料》由会员上传分享,免费在线阅读,更多相关内容在学术论文-天天文库。
1、LecturesontheGeometryofManifoldsLiviuI.NicolaescuJuly27,2007IntroductionShapeisafascinatingandintriguingsubjectwhichhasstimulatedtheimaginationofmanypeople.Itsu±cestolookaroundtobecomecurious.Eucliddidjustthatandcameupwiththe¯rstpurecreation.Relyingonthecommonexperience,hecreatedanabstractwo
2、rldthathadalifeofitsown.Asthehumanknowledgeprogressedsodidtheabilityofformulatingandansweringpenetratingquestions.Inparticular,mathematiciansstartedwonderingwhetherEuclid'sobvious"absolutepostulateswereindeedobviousand/orabsolute.ScientistsrealizedthatShapeandSpacearetwocloselyrelatedconcep
3、tsandaskedwhethertheyreallylookthewayoursensestellus.AsFelixKleinpointedoutinhisErlangenProgram,therearemanywaysoflookingatShapeandSpacesothatvariouspointsofviewmayproducedi®erentimages.Inparticular,themostbasicissueofmeasuringtheShape"cannothaveaclearcutanswer.ThisisabookaboutShape,Spacean
4、dsomeparticularwaysofstudyingthem.Sinceitsinception,thedi®erentialandintegralcalculusprovedtobeaveryversatiletoolindealingwithpreviouslyuntouchableproblems.ItdidnottakelonguntilitfoundusesingeometryinthehandsoftheGreatMasters.Thisisthepathwewanttofollowinthepresentbook.Intheearlydaysofgeomet
5、rynobodyworriedaboutthenaturalcontextinwhichthemethodsofcalculusfeelathome".Therewasnoneedtoaddressthisaspectsincefortheparticularproblemsstudiedthiswasanon-issue.Asmathematicsprogressedasawholetheaturalcontext"mentionedabovecrystallizedinthemindsofmathematiciansanditwasanotionsoimportant
6、thatithadtobegivenaname.Thegeometricobjectswhichcanbestudiedusingthemethodsofcalculuswerecalledsmoothmanifolds.Specialcasesofmanifoldsarethecurvesandthesurfacesandthesewerequitewellunderstood.B.Riemannwasthe¯rsttonotethatthelowdimensionalideasofhistimewereparticularaspectsofahigherdimensiona
7、lworld.The¯rstchapterofthisbookintroducesthereadertotheconceptofsmoothmanifoldthroughabstractde¯nitionsand,moreimportantly,throughmanywebelieverelevantexamples.Inparticular,weintroduceatthisearlystagethenotionofLiegroup.Themaingeometricandalgebraic
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