Lectures on Differential Geometry[Wulf Rossmann]

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1、LecturesonDifferentialGeometryWulfRossmann000(UpdatedOctober2003)2TothestudentThisisacollectionoflecturenoteswhichIputtogetherwhileteachingcoursesonmanifolds,tensoranalysis,anddifferentialgeometry.Iofferthemtoyouinthehopethattheymayhelpyou,andtocomplementthelectures.Thestyleisuneven,sometimespedantic,s

2、ometimessloppy,sometimestelegramstyle,sometimeslong–winded,etc.,dependingonmymoodwhenIwaswritingthoseparticularlines.Atleastthissetofnotesisvisiblyfinite.Thereareagreatmanymeticulousandvoluminousbookswrittenonthesubjectofthesenotesandthereisnopointofwritinganotheroneofthatkind.Afterall,wearetalkingab

3、outsomefairlyoldmathematics,stilluseful,evenessential,asatoolandstillfun,Ithink,atleastsomepartsofit.Acommentaboutthenatureofthesubject(elementarydifferentialgeometryandtensorcalculus)aspresentedinthesenotes.Iseeitasanaturalcontinuationofanalyticgeometryandcalculus.Itprovidessomebasicequipment,whichi

4、sindispensableinmanyareasofmathematics(e.g.analysis,topology,differentialequations,Liegroups)andphysics(e.g.classicalmechanics,generalrelativity,allkindsoffieldtheories).Ifyouwanttohaveanotherviewofthesubjectyoushouldbyallmeanslookaround,butIsuggestthatyoudon’tattempttouseothersourcestostraightenoutpr

5、oblemsyoumighthavewiththematerialhere.Itwouldprobablytakeyoumuchlongertofamiliarizeyourselfsufficientlywithanotherbooktogetyourquestionansweredthantoworkoutyourproblemonyourown.Eventhoughthesenotesarebrief,theyshouldbeunderstandabletoanybodywhoknowscalculusandlinearalgebratotheextentusuallyseeninsecon

6、d-yearcourses.Therearenodifficulttheoremshere;itisratheramatterofprovidingaframeworkforvariousknownconceptsandtheoremsinamoregeneralandmorenaturalsetting.Unfortunately,thisrequiresalargenumberofdefinitionsandconstructionswhichmaybehardtoswallowandevenhardertodigest.(Inthissubjectthedefinitionsaremuchhar

7、derthanthetheorems.)Inanycase,justbyrandomlyleafingthroughthenotesyouwillseemanycomplicatedlookingexpressions.Don’tbeintimidated:thisstuffiseasy.Whenyoulookedatacalculustextforthefirsttimeinyourlifeitpro