# riemannian geometry: a beginner's guide

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Frank MorganJones and Bartlett Publishers (1993)

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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'-' -'-' ,-' -' .U] UJ.'-C' ;. J.VJ. '-UJ. vaLUJ.'-.'T". . .,. ,-, ". .I.: UIV;:'l ,._,. , . ,." .uJ.an;:,_:-,' , '-'.LJ.vUJ. U] r lUOl\.-UJ. v OlU \.- ;:," ._,rLnv vau,-".. . . .., __" J. U"VI. "u.. LetIVl oe;)t"j.lY Wlln1 T' .l-Uf VUl"""4 V "'::> V .fr>... V r:;'r>.....,. c::. lLI .- ..........., _.r_r ..... .., ...... 1 ...... u ...... ,L,J..., ..,...", ......... ...... U ... ...,'y1-.. ... 1'P2 EM2 , identifyT= Tp1M1 = Tp2M2via a linear isometry. Let Bbe,4- (\ :'A 'T''C',y....""'AA 'C'n.... "" .. '1'1' ..wr ..r vw.... .r V H" .L vr .. ,."r ...... .L.J.n.pPl wr .. "".L.J.n.pP2 w,"""JJ .. vr....: ...1tA "" ... A 1tA T/>4- A l-./> "" ; ... D rw ... A I/>4- AI A l-.n ...yr ......JJI....J ....... v J. .... 1 w, .. "" J. .... L./ ........... , .... AJ, ..... ,' ......... /1' T,L, v ..... ......,images inMt, M2 Thenlength(11)< length(12)'T._1 : ..t. i\A i\A.11.... '"..t..", '" .1..1.11 -r1-, J.J'.L1 VI J.J'.L2LJ l.V V'"' U 1'""Euclidean space, or hyperbolic space, all of whichhave well-knowntrigonometries. Thus oneobtains distance estimates onthe other1..1 P 11I11111Uli 11111JUIIIIII:"o..In nature, the energy of a path or surface often depends on directionas well as length or area. The surface energy of a crystal, for example,depends radically on direction.Indeed, some directions are so muchcheaper that most crystals use onlyafewcheapdirections. (SeeFigure 10.1.) This chapter applies more general costs or norms tocurves and presents an appropriate generalization of curvature.convexity of is equivalent to the convexity of its unit ball{x:(x) \$ I}.For any curve C, parametrized by a differentiable mapy: [0, 1] Rn, define90n.",AT NnRMO;;: '-II///,"-// "/-r---.........-'- \. '"/ ('jJ ,/ ..,,./1"II I -IIIID ./1II I/ 1/,...,I,..,/ \. ,'"""'----............./.11\ "..1-.= .. h... ll it> 1"e'. .... ..' ............)convex, thereisalinear functionorI-form a VI..LI 1 .1..1",...... V" ,.,. . 1. T> 'T'1.auy \,.-Ul V 1.1 VIUf]. lVU. .1. (C')= J (T) ds> Jcp dsC' C'= Jcp ds= ( C)CbyStokes's Theorem, soC is-minimizing. Ifis strictly convex,theinequalityis strictunlessC1 isalsoastraightlinefromAtoB,soC is uniquely minimizing.10.3. Proposition. A nonnegativehomogeneous C2function onRnis convex (respectively, uniformly convex) if and only if therestrictions ( 8) of tocircles about theorigin satisfy"( 8) +( 8)< 0 0).rrooT. conveXIIVIIIeveryDIaneInrOUlmUISeaUIvalemTOconveXiTY wem::-lvassumen- L. I necurvaTure Korany - &r - TI til III notarCOnQIIlOnSISQIVenDV. .., . ..,f rr +Lf/ rL rIL\;jIL\J J J'T"t " ..1- ...1-.a. ........ ,",\.U ' ......u ... V.l.....LI (])I\ "\ / ffi2v -r..1 . ,1- t .11Y V.l..........l.l..v.....3\I fCD+CD' \ffi 12I ' /1/ffi/n\.a., -r\ v J'T'1. 1.. U..') ..lU.'I. lJenerallzeocurvamre. L el l Dea l curveWllnarclem!ln. . .')naramelrl7alIOnf: III I IK ana curvatureveccorK. 1 ettVne al. norm. Considervariations of supported in(0, 1). Thenthe first vari-ation satisfieso(f)= -JD2(K)'of ds,[0,1]YVHC;;I C;; 1J ' 1 c;;y,ItHe:::T 1ft /JUTII,I....C;;I.v..... ... THUH tA Ut Ute:::I r.1 r I ,I /'\rbLUJ - - I K bJas.Jro11 .,.. 111 wt: l:all U 'lKI LIlt: ;;;C;;HI: 'I"ll '-l.. ...., ' ........,f{ ,,\J \' /n- J \J /r (h{ .f'{,,\\A" fAr o:>n,T'\.;F,.I rl\1 ..\ J.r= - I D 2cp( (')( ("). 8((u) duJuy '-' uy pUJ.L". -.JUJ.'-''-' J.VJ. LU'-' UJ.'-' ,-,.r .J.. area SRemarks. For thecasewhere islengthandCoistheunitcircle, (1) saysIKI s; 1. The smoothness hypothesis onCo is unneces-sary; still the conclusionimplies that Co is Cl,l. If Co bounds aunique smooth area-minimizing surfaceSo with n the inward normalto Co along So, D2(K) actually must be a constant multiple ofn. Inparticular, aplanaroptimal isoperimetriccurvehasconstantgeneralizedcurvature:Proof. Let f:[0, a]Rnbe a local arc length parameterizationof Co. Consider compactly supported variationsSf. Theno:> o(area S- a( C)2):>-JISfl ds + 2a( Co) JD2( K) . ofdsby10.4. Therefore_______ 2a(Cu) 2 area Sol1.. norm is called crystalline if the unit-ball is polytope.Figure10.6. Length-minimizing networksmeet inthrees at equal angles of 120.10.9. Conjecture. Ifiscrystalline, thenanoptimal isoperimetriccurve is a polygon.10.10. cI'-minimizingnetworks. AnetworkNisafinitecollection10.12. Theorem [LM, Theorem 4.4]. Let be a differentiable normon Rn In -minimizingnetworks, n + 1segments can meet at apoint, but nevern + 2.It turnsout that all suchjunctionslocallycanbe"calibrated"and classified.GENERAL NORMS 99// / [,\./ /.':".,\./ /'"::. -::.\./ /': :: :::'. :\.//..... \. , ,// ': ... \. \\/F:. .: '" . :."1\ \/ /'... : : .... :\./. \ \r /\. . . .' .:'. '::.',::' ..,,' ,\. / \ \. . ../:\" , L' , ,.' .......0 . '.-" .:,':::' I \1\,I, e.... : . .....i E{:\;i,;,II\.,I,' I 1E::;:,::,:;i:.::::,',IV:11/;':.:1 "':'::;::::,',\':':::'\11-',:-/ \.',::::'::'::;;::';If)., II:;"\::,,!:.:;,:;11.1 \::':':::IU V,,::,II'\\\Figure 10.7. Soap filmsmeet in threesat1200anglesin anattempt tominimizearea.Thpnpvt'nul1th'T 1 (l 11-,, tn 'C -0 J.U.J.-'. J.UIl;UIIl;UI;,)lVl.t\.L_\YUfUUP, vvuuams, su1988, [A3, A4]). Consider piecewisedifferentiable, uniformlycon-vexnorms