Notes on Difierential Geometry - CMU

1 NotesonDi erentialGeometrywithspecialemphasisonsur facesinR3 MarkusDesernoMay 3, 2004 Department of ChemistryandBiochemistry, UCLA,LosAngeles,CA90095-1569,USAMax-Plan ck-Institutf ur Polymerforschung,Ackermannweg 10,55128 Mainz,GermanyThesenotesareanattemptto summarizesomeof thekeymathe-maticalaspectsof di erentialgeometry, as theyapplyin particularto thegeometryof surfacesinR3. Thefocusis notonmathematicalrigorbutratheroncollect ingsomebitsandpiecesof theverypow-erfulmachineryof manifoldsand\post-Newtoniancalculus".Eve nthoughtheultimategoalof eleganceis a completecoordinatefreedescription,thisgo alis farfrombeingachievedhere|notbecausesuch a descriptiondoes notexistyet,butbecausetheauthoris farto thegeometricaspectsaretakenfromFrankel's book[9],onwhich thesenotesrelyheavily. For \classical"di erentialgeometryof curves andsurfacesKreyszigbook[14] hasalsobeentakenas a presentationvariesquitea greatdetail,othersareonlytoucheduponquic kly, mostlywiththeintent to indicateintowhich directiona particularsubjectmight be thetheory of nitions.

2 Thesurface.. WeingartenandGauss.. conditions.. Identities.. nitionandproperties.. termsofrk.. expansion.. : Arclengthparameterization.. : Height is a functionof axialdistance.. : Axialdistanceis a functionof height .. a nitionof thevariation.. the rstfundamentalform.. of themetric.. thenormalvector.. thevolume.. theextrinsicgeometry.. problemsinvolvingthe .. ningproperty .. : Soap lmbetweentwo circles.. : Helicoid.. : Enneper'sminimalsurface.. 'sformula.. analysisfortheisoperimetricproblem.. equation.. of freecylindricalvesicles.. nitionandtransformationlaw.. coordinates.. erentialsandandpull-backs.. elds.. of Killing elds.. ,paralleltransport andcovariantdi .. of Levi-Civit a.. di erentiation.

3 :ThePoincar e plane.. elsymbols.. eldsof thePoincar e metric.. of a function,i. e., a scalar.. of a vector eld.. fora 1-form.. of a generaltensor eld.. :Liederivative of themetric.. problems62 Bibliography6431. Somefundamentalsof thetheory of thesurfaceLetUbe an(open)subsetofR2andde nethefunction~r: R2 U!R3(u1;u2)7!~r(u1;u2):( )We willassumethatallcomponents of thisfunctionaresu cientlyoftendi nefurtherthevectors1e ~r; :=@~r@u ;( )and~n:=e1 e2je1 e2j:( )If thee areeverywherelinearlyindependent2, themapping( )de nesa a di erentiablesubmanifoldofR3. Thevectorse (~r) belongtoT~rS, thetangent spaceofSat~r, thisis why weusea di erent notationforthemthanthe\ordinary"vectorsf romR3. Notethatwhile~nis a unitvector,thee aregenerallynotof rstfundamentalformonthesurfaceSis de nedasgij:=ei ej:( )It is a secondranktensorandit is it is furthermore(everywhere)diagonal, thecoordinatesarecalledlocally denotedasgij, so thatwe havegijgjk= ki= 1ifi=k0ifi6=k;( )where kiis calledtheKroneckersymbol.

4 Hence,thecomponents of theinversemetricaregivenby g11g12g21g22 =1g g22 g21 g12g11 :( )Byvirtueof Eqn.( )themetrictensorcanbe usedto raiseandlower indicesin ,\indicesupor down"meansthatwe arereferringto components of tensorswhich live in thetangent spaceor thecotangent space ,respectively. It requirestheadditionalstructureof a metricin themanifoldin orderto de neanisomorphismbetweenthesetwo di erent of the rstfundamentalformis givenbyg:=detg jgj jgijj=12"ik"jlgijgkl;( )where"ikis thetwo-dimensionalantisymmetricLevi-Civi t a symbol"ik= i1 i2 k1 k2 = i1 k2 k1 i2; "ik="ik:1e =@~r=@u is @=@u (oreven shorter:@u ) thecanonicallocalcoordinatebasisbelongin gto requirement is thatthedi erential~r hasrank2 ( ).4ntt SClocal tangent :Illustrationofthedef-initionofthenormal curvature n,Eqn.

5 ( ),andthegeodesiccurva-ture g, Eqn.( ).Theyareessen-tiallygiven by theprojectionof_~tontothelocalnormalvect orandonto thelocaltangent plane, 'is theanglebetweene1ande2, thenwe haveje1 e2j2=je1j2je2j2sin2'=g11g22(1 cos2') =g11g22 (e1 e2)2=g11g22 g12g21=g :Hence,we haveje1 e2j=pg somecurveCde nedonthesurfaceS, which goes throughsomepointP, at which thecurvehasthetangent vector ~tandprincipalnormalvector~p=_~t= , andat which point thesurfacehasthenormalvector~n|seeas now have thefollowingtwo equations:~p ~n=cos#and_~t= ~p :The rstde nestheangle#betweenthetwo unitvectors~nand~p, thesecondde nesthecurvatureof ,we obtain cos#=_~t ~n :( )If thecurve is parameterizedasui(s), we have_~t(s) = ~r(s) =@2@s2~r(s) =@@s ~r;i_ui =~r;ij_ui_uj+~r;i ui=ei;j_ui_uj+ei ui:Sinceei ~n= 0, we obtainfromthisandEqn.

6 ( ) cos#=_~t ~n= ei;j ~n _ui_uj:( )Theexpressionin bracketsis independent of thecurve anda property of is calledthesecondfundamentalform,andwe willtermitbij:bij:=ei;j ~n :( )Sinceei;j=ej;i, thesecondfundamentalformis symmetricin itstwo thesecondfundamentalformisfurthermoredia gonal, rstandsecondfundamentalformarediagonal,t hecoordinatelinesareorthogonalandtheyfor mlinesof curvature,i. e., theylocallycoincidewiththeprincipaldirec tionsof curvature(seebelow).Di erentiatingtheobviousrelationei ~n= 0 withrespecttoujshowsthatei;j ~n+ei ~n;j= 0, fromwhich followsthatthesecondfundamentalformis alsogivenbybij:= ei ~n;j:( )Thisexpressionis usuallylessconvenient, sinceit involves thederivative of aunitvector,andthus thederivativeof square-root ,thecurvature of a curve at thepointPis partiallydueto thefactthatthecurve itselfiscurved,andpartiallybecausethesur faceis orderto somehow disentanglethesetwo e ects,it it usefulto de nethetwo conceptsnormalcurvatureandgeodesiccurvat ure.

7 We follow Kreyszig[14] in [14, paragraph60]fora moredetaileddiscussiononwhatthisimplies5 Thelefthandsideof Eqn.( )onlydependsonthedirectionof thecurve atP,i. e.~t, ,it is actuallya property of thesurface. It is calledthenormalcurvature nof thesurface in thedirection~t. If we performa reparameterizationof thecurve, we nd_ui= (dui=dt)(dt=ds) =u0i=s0, andfromthatwe nd: n:= cos#=biju0iu0jgiju0iu0j=bijduidujgijduid uj:( )Thenormalcurvatureis thereforetheratiobetweenthesecondandthe ( )showsthatthenormalcurvatureis a quadraticformof the_ui, or looselyspeakinga quadraticformof thetangent is thereforenotnecessaryto describe thecurvaturepropertiesof asurfaceat everypoint by givingallnormalcurvaturesin is enoughto know is naturalto ask,in which directionsthenormalcurvatureis ( )as bij ngij vivj=0;anddi erentiatingthisexpressionwithrespecttovk (treating nasa constant, sinced n= 0 is a necessaryconditionfor nto be extremal),we nd bik ngik vi=0.

8 Or afterraisingtheindexk bki n ki vi=0:( )Thisis an important result:It showsthatthesearch forextremalcurvaturesandthecorresponding directionsleadsto aneigenvalueproblem: Thedirectionsalongwhich thenormalcurvatureis extremalaregiven by theeigenvectorsof thematrixbki, eigenvaluesarecalledprincipal curvatures, andwe willcallthem 1and 2. Thispermitsusto de nethefollowingtwo importantconcepts:MeancurvatureHandGauss iancurvatureKarede nedas sumandproductof theprincipalcurvatures2H:= 1+ 2=bii;( )K:= 1 2= bki = bijgjk = bij gjk =bg;( )wherebis thedeterminant of thesecondfundamentalform:b:=detb jbj jbijj=12"ik"jlbijbkl;( )Sincethede nitionsofHandKinvolve theeigenvaluesofbji, theyareinvariant underreparametrizationsof thesurfacebendsin space ,theso calledgeodesiccurvature gis ameasureof how a curve curves ona surface,which is independent of thecurvatureof obtainedby projectingthevector_~tof thecurve onto thelocalnormalvectorof thesurface,thegeodesiccurvatureis obtainedby projecting_~tonto thelocaltangent plane,thereby essentiallyprojectingoutany curvaturedeformationsof ,andby a similarargument as theonewhich leadto Eqn.

9 ( ),we nd g= sin# :( ) WeingartenandGaussA keyresultin thetheoryof spacecurves aretheformulasof Frenet, which expressthechangeof thelocalcoordinatesystem(tangent vector ,normalvector,andbinormalvector)up onmovements alongthecurve in termsof thisin thetheoryof surfacesaretheformulasby WeingartenandGauss,which describe thevariationof thelocalcoordinatesystemuponsmallmovemen ts ~n ~n= 1, di erentiationwithrespecttou gives~n; ~n= normalvectorupon(in nitesimally)movingonthesurfaceis parallelto canhencebe expressedas a linearcombinationof thetangent vectors,i. e., we canwrite~n; =A e . A scalarmultiplicationwithe togetherwithEqn.( )showsthatA = b , andwe thereby obtaintheformulaof Weingarten:~n; = b e (Weingarten):( )Thechangeof thetangent vectorsis generallyalongallthreedirectionsof thelocalcoordinatesystem,so wemay writee ; =A e +B ~n.

10 A scalarmultiplicationwith~ntogetherwithEq n.( )immediatelyshowsthat6B =b . A scalarmultiplicationwithe showsthatA =e ; e , wherewe de nedA =A g . NotethatthisshowsthatA is symmetricin its rsttwo indices,becausee ; =~r; =~r; .Letus now lookat derivatives of themetric:g ; =(e e ); =e ; e +e e ; :TogetherwiththeexpressionforA andaftercyclicpermutation,we obtainthethreeequivalent equations:g ; =A +A ;( )g ; =A +A ;( )g ; =A +A :( )If we now addthe rsttwo of theseequationsandfromthatsubtractthethir done,i. e., if we formthecombination( )+( ) ( ),andadditionallyexploitthesymmetryofA in its rsttwo indices,we ndA =12 g ; +g ; g ; :ThisshowsthattheA aresimplytheChristo elsymbolsof the rstkind|seeEqn.( ).Andhence,A aretheChristo elsymbolsof thesecondkind|seeEqn.