# Riemannian Geometry: a Beginner's Guide

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Frank MorganJones and Bartlett Publishers (1993)

### Text of Riemannian Geometry: a Beginner's Guide

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hl"-'OTIhlp. I(T:\l1"" '"1"- :\11 '"Anm- lnK h:\" m f'l1TV:\-'" .. .ITP."- 1Ulll ... \..ii VVvl U I' r D.. D' 1- 'Jk'U''' "u'" ........"''''.. "' ................... '" ...... '" 61. "'Uv JR. = ),R... (7)J' 'J,iYI' ... . I' ,.11 yuulllIllll.. UIfiijkl i:1:S i:1 Illi:1l11XUIIi'! , , ,Ll\i1k1J Ll\i1k2J Ll\i1kmJ,[Rimkd [Rimk2] [Rimkm]thenR;1isthecorresnondim!matrixoftraces sothedefinitionofRic as a bilinear formdoes not really denend on the choice oforthonormal coordinates forTvS. Its application to el yields the sum .1. .1 .1V.l U.lV VU.lV V.l UA.li:> 1. o R 1(' (0. \ - fl - ).fl .. - ). fl ..1 , Ij '11-'Ul-IIIIi i401m.......=2.. K(el /\ eJ.I 'L.r""[JA D'T''CD .. ..Henceforany orthonormal basisVI, .., Vrn for TpS,rnVI . Ric (VI) = K(VI /\ Vi)' (8)i=2and foranyunit V ETpS, r",.1 I 9. vKIC lUI I 1\.1U/\ WI. IIunl -" IW.LU 'T"1. .1. n' 1. .J:.Luu.., ..u"" .L.......ua", au a", au'0' sectional curvatures.The scalar curvature R is defined as the trace of the Riccicurvature:R = (10)iHenceforany orthonormal basis v, . .... Vn>for TnS.'" mlm 11 IJ< L L KIll 1\ IJ:I I KIl"l I I I 1.,,,1(j])I -J JrE'&'. t17> 1. ... .J: .11 " 'T' C' 'T"1. ..1._..., ..u"" "''''' .. au6.- .L lJu, .L UU.., ..U"" "a .. :n_..... 1..J: .11 .1 .., rr..v .. U"" a -0 VLau ,,",UL 'a ..a.. a yvu....Remark. Historically Ric used to have the opposite sign. Sometexts givetheRiemannian curvaturetensor Rijkl the opposite sign.5.3. The covariant derivative. Let S be a C2m-dimensional surfacein.K 1I T isaoirreremiaDJeruncrionon.'\ menmeoerivalive vU. . .IS a veCIOrnelO. nUl III IS a veCIOrnelO lor a nelO or malflces. . .or 1 LJUU.Wl:SCInlnen (ne\. .val.lVe .vWIll nave. lO.:>. Int: InlO1 ISlnt:. - --

unt u.. L ...."" L""" .. 'r '--' _.....1.. ...... ' _1. ,11' 1 .. u.., .. ..,..,U uu.., ..... u .......,.A y"tn h r 1that is, a geodesic forthe standard metricds2=dx 2+ dy 2 + dz 2+ dt2 (1)It is actuallya geodesic forany metric of the form..1_2...LA. 2...L..1_2...L..1.2'1\""..,"1"""" "L "".r"4""\ ...,........ . . . . .It:li:1l1VllYUIl lWU1 'T't. ... 1 .1... 11 . 1 i'.&......"" ..u ',," 'V.."J,,"'V'V"" L"'" ""u.. ...u... .. .. L,u.."",," 'V....1.. .......... 11 n.:"t.UU"'L .. .." L'V.. 'V... ..,..,.. .., ....,'0,......velocity relative to one another. (Of course, in acceleratingreferenceframes, thelawsofphysicslookdifferent. Cupsof lemonade in acceleratingcars suddenlyfall over, andtennis balls onthefloors of rockets flattenlike pancakes.)2. The speed of a light beam is the same relative to any inertialframe, whether moving in the same or the opposite di-{thil1. , orr ".I ..., or '0thp h" thp"0 r .I-It toothpr.I r1;;.1ll'htlmp'l;;. I;;. rlown hH1h \'-' '-',Einstein'spostulatesholdformotionalonggeodesics in space-time if onetakes the special case of (2):ds2=-dx2- dy 2 -dz 2+ c2dt2 (3),..n,1/'\ rHA1>TPR7 thp. I w1thr thp. of l10ht WP. w111. '-'tA r 1Thp. I of'-'hllt IAAin-.I'-''.1....or np.w ot thp. ot.thp nAt tAr thpnAvpl0'r rthat thesquare of the length of curves in space-time can be positiveor negative, all definitions and properties remain formally the same.Inparticular, positivesectional curvaturemeansthat parallel geo-desics converge (the square of the distance between them decreases).(See Section 6.7.)This newdistance s is oftencalled"proper time" T, since amotionless particle (x, y, Zconstant) has ds2=dt2Ifwereplacethesymbol s byT and changetospherical coordinates, theLorentzmetric becomesdT2= -dr2- r2dq/ - r2sin2

'"'T""'''' th"t fLt. .... ? ....... J J"leo: nv V V 1'TT2/l2.Assumeyis parameterized by arc length t. Let Wbean ortho-GEODESICSANDGLOBAL GEOMETRY 87gonal, parallel unit vectorfield on'Y.Take as a variational vectorfieldI ".,. \(sin J t )W, which vanishes at theendpoints of 1'. By (4), the initial\ I-' , ,VI.-0I;' uy[2 ,.I I 7T 7T\ I . ,,7T \ - 1 cos 1 t } sm 1 t }1, W)

0I7T227T.27T . 27TA----,12 \.-v;:' 1 12 ;:,IU 1 V.0Thic.... , .. 'nfthp .,.nf ,,\, 'JOC 'JOJn'JOth.thpnrnnfI r--rT> JTn thor ___ f n'e- ,.,J."70 ,,___ ..lA h.,.,or,-

-.,.n ... " ..... ;f- .'- ,TTIT nf-... f- ...L'-'L.. .....0 of l' (extendingWbyparallel transport). Averagingover all suchchoices permits us toreplacethe boundonKbyaboundonitsaverage, theRicci curvature. Thetheorem of Myers concludes thatif Ric> (n- l)Ko, then diam M< 7T/VKo.9.6. Constant curvature, the Sphere Theorem, and the Rauch Com-parisonTheorem. Thissection justmentionssomefamousresultson a smooth,connected,completeRiemannian manifold M.Suppose that M is simply connected, so every loop can be shrunkto a point. Suppose the sectional curvature Kis constant for allsections at all points. By scaling, wemay assumeKis1, 0, or -l.If K=1,Mis theunit sphere. If K=0, Mis Euclidean space.T T7 .. -"1' ,, ,.'T'1. .'-..J ,,.H..n. .I., H.I. "U]P'-.I.VVU'- "pa,-,- .I.UU" u'- auu'-' -'-'

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