现代几何学方法和应用

Upuntilrecently,Riemanniangeometryandbasictopologywerenotincluded,evenbydepartmentsorfacultiesofmathematics,ascompulsorysubjectsinauniversity-levelmathematicaleducation.Thestandardcoursesintheclassicaldifferentialgeometryofcurvesandsurfaceswhichweregiveninstead(andstillaregiveninsomeplaces)graduallycametobeviewedasanachronisms.However,therehasbeenhithertonounanimousagreementastoexactlyhowsuchcoursesshouldbebroughtuptodate,thatistosay,whichpartsofmoderngeometryshouldberegardedasabsolutelyessentialtoamodernmathematicaleducation,andwhatmightbetheappropriatelevelofabstractnessoftheirexpositionThetaskofdesigningamodernizedcourseingeometrywasbegunin1971inthemechanicsdivisionoftheFacultyofMechanicsandMathematicsofMoscowStateUniversity.Thesubject-matterandlevelofabstractnessofitsexpositionweredictatedbytheviewthat,inadditiontothegeometry,ofcurvesandsurfaces,thefollowingtopicsarecertainlyusefulinthevariousareasofapplicationofmathematics(especiallyinelasticityandrelativity,tonamebuttwo),andarethereforeessential:thetheoryoftensors(includingeovariantdifferentiationofthem);Riemanniancurvature;