luke g., mishchenko a. vector bundles and their applications (kluwer, 2000)(262s)_mdat_

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1、VECTORBUNDLESANDTHEIRAPPLICATIONSVECTORBUNDLESANDTHEIRAPPLICATIONSGlenysLUKEOxfordUniversityOxfordAlexanderMISHCHENKOMoscowStateUniversityMoscowKLUWERACADEMICPUBLISHERSBoston/London/DordrechtCONTENTSvPREFACEInthelastfewyearstheuseofgeometricmethodshaspermeatedmanymorebranchesofmathematicsand

2、thesciences.Brieflyitsrolemaybecharacterizedasfollows.Whereasmethodsofmathematicalanalysisdescribephenomena‘inthesmall’,geometricmethodscontributetogivingthepicture‘inthelarge’.Asecondnolessimportantpropertyofgeometricmethodsistheconvenienceofusingitslanguagetodescribeandgivequalitativeexplan

3、ationsfordiversemathematicalphenomenaandpatterns.Fromthispointofview,thetheoryofvectorbundlestogetherwithmathematicalanalysisonmanifolds(globalanal-ysisanddifferentialgeometry)hasprovidedamajorstimulus.Itslanguageturnedouttobeextremelyfruitful:connectionsonprincipalvectorbundles(intermsofwhic

4、hvariousfieldtheoriesaredescribed),transformationgroupsincludingthevarioussymmetrygroupsthatariseinconnectionwithphysicalproblems,inasymptoticmethodsofpartialdifferentialequationswithsmallparameter,inellipticoperatortheory,inmathematicalmethodsofclassicalmechanicsandinmathematicalmethodsinecon

5、omics.Thereareothercur-rentlylesssignificantapplicationsinotherfields.Overasimilarperiod,uni-versityeducationhaschangedconsiderablywiththeappearanceofnewcoursesondifferentialgeometryandtopology.Newtextbookshavebeenpublishedbut‘geometryandtopology’hasnot,inouropinion,beenwellcoveredfromaprac-tic

6、alapplicationspointofview.Existingmonographsonvectorbundleshavebeenmainlyofapurelytheoreticalnature,devotedtotheinternalgeometricandtopologicalproblemsofthesubject.Studentsfromrelateddisciplineshavefoundthetextsdifficulttouse.Itthereforeseemsexpedienttohaveasimplerbookcontainingnumerousillustr

7、ationsandapplicationstovariousproblemsinmathematicsandthesciences.Partofthisbookisbasedonmaterialcontainedinlecturesoftheauthor,A.Mishchenko,giventostudentsoftheMathematicsDepartmentatMoscowStateUniversityandisarevisedversionoftheRussianeditionof19