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1、Semi-RiemannGeometryandGeneralRelativityShlomoSternbergSeptember24,200320.1IntroductionThisbookrepresentscoursenotesforaonesemestercourseattheundergraduatelevelgivinganintroductiontoRiemanniangeometryanditsprincipalphysicalapplication,Einstein’stheoryofgeneralrelativity.Thebackgroundassum

2、edisagoodgroundinginlinearalgebraandinadvancedcalculus,preferablyinthelanguageofdi?erentialforms.ChapterIintroducesthevariouscurvaturesassociatedtoahypersurfaceembeddedinEuclideanspace,motivatedbytheformulaforthevolumefortheregionobtainedbythickeningthehypersurfaceononeside.Ifwethickenthe

3、hypersurfacebyanamounthinthenormaldirection,thisformulaisapolynomialinhwhosecoe?cientsareintegralsoverthehypersurfaceoflocalexpressions.Theselocalexpressionsareelementarysymmetricpolynomialsinwhatareknownastheprincipalcurvatures.Theprecisede?nitionsaregiveninthetext.Thechapterculminateswi

4、thGauss’Theoremaegregiumwhichassertsthatifwethickenatwodimensionalsurfaceevenlyonbothsides,thenthetheseintegrandsdependonlyontheintrinsicgeometryofthesurface,andnotonhowthesurfaceisembedded.Wegivetwoproofsofthisimportanttheorem.(Wegiveseveralmorelaterinthebook.)The?rstproofmakesuseof“norm

5、alcoor-dinates”whichbecomesoimportantinRiemanniangeometryand,as“inertialframes,”ingeneralrelativity.ItwasthistheoremofGauss,andparticularlytheverynotionof“intrinsicgeometry”,whichinspiredRiemanntodevelophisgeometry.ChapterIIisarapidreviewofthedi?erentialandintegralcalculusonman-ifolds,inc

6、ludingdi?erentialforms,thedoperator,andStokes’theorem.Alsovector?eldsandLiederivatives.Attheendofthechapterareaseriesofsec-tionsinexerciseformwhichleadtothenotionofparalleltransportofavectoralongacurveonaembeddedsurfaceasbeingassociatedwiththe“rollingofthesurfaceonaplanealongthecurve”.Cha

7、pterIIIdiscussesthefundamentalnotionsoflinearconnectionsandtheircurvatures,andalsoCartan’smethodofcalculatingcurvatureusingframe?eldsanddi?erentialforms.WeshowthatthegeodesicsonaLiegroupequippedwithabi-invariantmetricarethetranslatesoftheoneparametersubgroups.Ashort