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1、IntroductiontoDifferentialGeometry&GeneralRelativityThiiirdPriiintiiingJanuary2002LectureNotesbyStefanWanerwithaSpecialllGuestLecturebyGregoryC..LevineDepartmentsofMathematicsandPhysics,HofstraUniversityIntroductiontoDifferentialGeometryandGeneralRelativityLectureNotesbyStefanWaner,witha
2、SpecialGuestLecturebyGregoryC.LevineDepartmentofMathematics,HofstraUniversityThesenotesarededicatedtothememoryofHannoRund.TABLEOFCONTENTS1.Preliminaries:Distance,OpenSets,ParametricSurfacesandSmoothFunctions2.SmoothManifoldsandScalarFields3.TangentVectorsandtheTangentSpace4.Contravariant
3、andCovariantVectorFields5.TensorFields6.RiemannianManifolds7.LocallyMinkowskianManifolds:AnIntroductiontoRelativity8.CovariantDifferentiation9.GeodesicsandLocalInertialFrames10.TheRiemannCurvatureTensor11.ALittleMoreRelativity:ComovingFramesandProperTime12.TheStressTensorandtheRelativist
4、icStress-EnergyTensor13.TwoBasicPremisesofGeneralRelativity14.TheEinsteinFieldEquationsandDerivationofNewton'sLaw15.TheSchwarzschildMetricandEventHorizons16.WhiteDwarfs,NeutronStarsandBlackHoles,byGregoryC.Levine21.PreliminariesDistanceandOpenSetsHere,wedojustenoughtopologysoastobeableto
5、talkaboutsmoothmanifolds.Webeginwithn-dimensionalEuclideanspaceEn={(y1,y2,...,yn)
6、yiéR}.Thus,E1isjusttherealline,E2istheEuclideanplane,andE3is3-dimensionalEuclideanspace.Themagnitude,ornorm,
7、
8、yy
9、
10、ofy=(y1,y2,...,yn)inEnisdefinedtobe222
11、
12、yy
13、
14、=y1+y2+...+yn,whichwethinkofasitsdistanc
15、efromtheorigin.Thus,thedistancebetweentwopointsy=(y1,y2,...,yn)andz=(z1,z2,...,zn)inEnisdefinedasthenormofz-yy:DistanceFormula222Distancebetweenyandz=
16、
17、zz-y
18、
19、=(z1-y1)+(z2-y2)+...+(zn-yn).Proposition1.1(Propertiesofthenorm)Thenormsatisfiesthefollowing:(a)
20、
21、yy
22、
23、≥0,and
24、
25、yy
26、
27、=0
28、iffy=0(positivedefinite)(b)
29、
30、¬yy
31、
32、=
33、¬
34、
35、
36、yy
37、
38、forevery¬éRandyéEn.(c)
39、
40、yy+zz
41、
42、≤
43、
44、yy
45、
46、+
47、
48、zz
49、
50、foreveryy,zéEn(triangleinequality1)(d)
51、
52、yy-zz
53、
54、≤
55、
56、yy-w
57、
58、+
59、
60、ww-z
61、
62、foreveryy,,,z,,,wéEn(triangleinequality2)TheproofofProposition1.1isanexercisewhichmayrequirereferencetoalinearalgebrat
