Lectures on Differential Geometry-Math 240BC

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1、LecturesonDierentialGeometryMath240BCJohnDouglasMooreDepartmentofMathematicsUniversityofCaliforniaSantaBarbara,CA,USA93106e-mail:moore@math.ucsb.eduJune5,2009PrefaceThisisasetoflecturenotesforthecourseMath240BCgivenduringtheWinterandSpringof2009.Thenote

2、sevolvedasthecourseprogressedandarestillsomewhatrough,butwehopetheyarehelpful.Starredsectionsrepresentdigressionsarelesscentraltothecoresubjectmatterofthecourseandcanbeomittedonarstreading.OurgoalwastopresentthekeyideasofRiemanniangeometryuptothegeneral

3、izedGauss-BonnetTheorem.TherstchapterprovidesthefoundationalresultsforRiemanniangeometry.ThesecondchapterprovidesanintroductiontodeRhamcohomology,whichprovidesprehapsthesimplestintroductiontothenotionofhomologyandcohomologythatissopervasiveinmoderngeome

4、tryandtopology.InthethirdchapterweprovidesomeofthebasictheoremrelatingthecurvaturetothetopologyofaRiemannianmanifold

5、theideahereistodevelopsomeintuitionforcurvature.FinallyinthefourthchapterwedescribeCartan'smethodofmovingframesandfocusonitsapplicationto

6、oneofthekeytheoremsinRiemanniangeometry,thegeneralizedGauss-BonnetTheorem.Thelastchapterismoreadvancedinnatureandnotusuallytreatedintherst-yeardierentialgeometrycourse.Itprovidesanintroductiontothetheoryofcharacteristicclasses,explaininghowthesecouldbe

7、generatedbylookingforextensionsofthegeneralizedGauss-BonnetTheorem,anddescribesapplicationsofcharacteristicclassestotheAtiyah-SingerIndexTheoremandtotheexistenceofexoticdierentiablestructuresonseven-spheres.iContents1Riemanniangeometry21.1Reviewoftangen

8、tandcotangentspaces..............21.2Riemannianmetrics.........................51.3Geodesics...............................91.3.1Smoothpaths.........................91.3.2Piecewisesmoothpaths...................131.4Hamilton'sprinciple....................

9、.....141.5TheLevi-Civitaconnection.....................201.6FirstvariationofJ..........................241.7Lorentzmanifolds...........................271.8TheRiemann-Christoelcurvaturetensor.............301.9Curvaturesym