4JohnMSullivanWithrespecttotheL2innerproducthUViRUpVpdAonvectoreldsthevectormeancurvatureisthenegativegradientoftheareafunctionaloftencalledtherstvariationofareaHrAreaSimilarlythenegat ID: 334156
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2JohnM.SullivanAplanecurveiscompletelydetermined(uptorigidmotion)byits(signed)curva-ture(s)asafunctionofarclengths.Foraspacecurve,however,weneedtolookatthethird-orderinvariants;thesearethetorsion andthederivative0,butthelatterofcoursegivesnonewinformation.Thesearenowacompletesetofinvariants:aspacecurveisdeterminedby(s)and (s).Genericallyspeaking,whilesecond-ordercurvaturessufcetodetermineahyper-surface(ofcodimension1),higher-orderinvariantsareneededforhighercodimension.ForcurvesinEd,forinstance,weneedd�1generalizedcurvatures,oforderupton,tocharacterizetheshape.Letusexaminethecaseofspacecurves E3inmoredetail.Ateverypointp2 wehaveasplittingofthetangentspaceTpE3intothetangentlineTp andthenormalplane.Aframingalong isasmoothchoiceofaunitnormalvectorN1,whichisthencompletedtotheorientedorthonormalframe(T;N1;N2)forE3,whereN2=TN1.Takingthederivativewithrespecttoarclength,wegetaskew-symmetricmatrix(aninnitesimalrotation)describinghowtheframechanges:0@TN1N21A0=0@012�10 �kappa2� 01A0@TN1N21A:Here,T0(s)=PiNiisthecurvaturevectorof ,while measuresthetwistingofthisframing.IfwechooseN1atabasepointalong anaturalchoiceofframingistheparallelframingorBishopframing[Bis75]denedbythecondition =0.Equivalently,thevectorsNiareparallel-transportedalong fromthebasepoint,usingtheRiemannianconnectioninducedbytheimmersioninE3.Oneshouldnotethatthisisnotnecessarilyaclosedframingalongaclosedloop ;whenwereturntothebasepoint,N1hasbeenrotatedthroughananglecalledthewritheof .Otherframingsarealsooftenuseful.Forinstance,if liesonasurfaceMwithunitnormal,itisnaturaltochooseN1=.ThenN2=:=Tiscalledthecornormalvector,and(T;;)istheDarbouxframe(adaptedto ME3).Thecurvaturevectorof decomposesintopartstangentandnormaltoMasT0=g+n.Here,nmeasuresthenormalcurvatureofMinthedirectionT,andisindependentof ,whileg,thegeodesiccurvatureof inM,isanintrinsicnotion,unchangedifweisometricallydeformtheimmersionofMintospace.Whenthecurvaturevectorof nevervanishes,wecanwriteitasT0=Nwhere�0andNisaunitvector,theprincipalnormal.ThisyieldstheorthonormalFrenetframe(T;N;B),whosetwisting isthetorsionof .ThetotalcurvatureofasmoothcurveisRds.In[Sul06]weinvestigatedanumberofresults:Forclosedcurves,thetotalcurvatureisatleast2(Fenchel)andforknottedspacecurvesthetotalcurvatureisatleast4(F´ary/Milnor).Forplanecurves,wecanconsiderinsteadthesignedcurvature,andndthatRdsisalwaysanintegralmultipleof2.Then(followingMilnor)wedenedthetotalcurvatureofapolygonalcurvesim-plytobethesumoftheturninganglesatthevertices.Thenallthesetheoremsontotal 4JohnM.SullivanWithrespecttotheL2innerproducthU;Vi:=RUpVpdAonvectorelds,thevectormeancurvatureisthenegativegradientoftheareafunctional,oftencalledtherstvaria-tionofarea:H=�rArea.(Similarly,thenegativegradientoflengthforacurveisitscurvaturevectorN.)Justasisthegeometricversionofsecondderivativeforcurves,meancurvatureisthegeometricversionoftheLaplacian.Indeed,ifasurfaceMiswrittenlocallyasthegraphofaheightfunctionfoveritstangentplaneTpMthenH(p)=f.Alternatively,wecanwriteH=rM=Mx,wherexisthepositionvectorinR3andMisBeltrami'ssurfaceLaplacian.Ifweowacurveorsurfacetoreduceitslengthorarea,byfollowingthesegradientsNandH,theresultingparabolicheatowisslightlynonlinearinanaturalgeomet-ricway.Thisso-calledmean-curvatureowhasbeenextensivelystudiedasageometricsmoothingow.3.IntegralCurvatureRelationsforSurfacesForsurfaces,theintegralcurvaturerelationswewanttoconsiderrelateareaintegralsoveraregionDMtoarclengthintegralsovertheboundary =@D.TheGauß/Bonnettheoremsays,whenDisadisk,2�ZZDKdA=I gds=I T0ds=�I0dx;wheredx=Tdsisthevectorlineelementalong .ThisimpliesthatthetotalGaußcurvatureofDdependsonlyonacollarneighborhoodof :ifwemakeanymodicationtoDsupportedawayfromtheboundary,thetotalcurvatureisunchanged(aslongasDremainstopologicallyadisk).WewillextendthenotionofGaußcurvaturefromsmoothsurfacestomoregeneralsurfaces(inparticularpolyhedralsurfaces)byrequiringthispropertytoremaintrue.TheotherrelationsareallprovedbyStokes'Theorem,andthusonlydependon beingtheboundaryofDinahomologicalsense;fortheseDisnotrequiredtobeadisk.FirstconsiderthevectorareaA :=1 2I xdx=ZZDdA:Theright-handsiderepresentsthetotalvectorareaofanysurfacespanning ,andtherelationshowsthistodependonlyon (andthistimenotevenonacollarneighborhood).Theintegrandontheleft-handsidedependsonachoiceoforiginforthecoordinates,butbecauseweintegrateoveraclosedloop,theintegralisindependentofthischoice.Bothsidesofthisvectorareaformulacanbeinterpreteddirectlyforapolyhedralsurface,andtheequationremainstrueinthatcase.WenotealsothatthisvectorareaA ispreservedwhen evolvesundertheHasimotoorsmoke-ringow. 6JohnM.Sullivan(SimilarargumentsleadtoanotionofGaußcurvature—denedasameasure—foranyrectiablesurface.Forourpolyhedralsurface,thismeasureconsistsofpointmassesatvertices.Surfacescanalsobebuiltfromintrinsicallyatpiecesjoinedalongcurvededges.TheirGaußcurvatureisspreadoutwithalineardensityalongtheseedges.Thistechniqueisoftenusedindesigningclothes,wherecornerswouldbeundesirable.)NotethatKpisclearlyanintrinsicnotion,asitshouldbe,dependingonlyontheanglesofeachtriangleandnotonthepreciseembeddingintoR3.Sometimesitisusefultohaveanotionofcombinatorialcurvature,independentofallgeometricinformation.Givenjustacombinatorialtriangulation,wecanpretendthateachtriangleisequilateralwithangles=60,whetherornotthatgeometrycouldbeembeddedinspace.TheresultingcombinatorialcurvatureisKp= 3(6�degp).Inthiscontext,theglobalformPKp=2 (M)ofGauß/BonnetamountstonothingmorethanthedenitionoftheEulercharacteristic .4.2.VectorareaThevectorareaformulaA :=1 2I xdx=ZZDdAneedsnospecialinterpretationfordiscretesurfaces:bothsidesoftheequationmakesensedirectly,sincethesurfacenormaliswell-denedalmosteverywhere.However,itisworthinterpretingthisformulaforthecasewhenDisthestarofavertexp.Moregener-ally,suppose isanyclosedcurve(smoothorpolygonal),andDistheconefrompto (theunionofalllinesegmentspqforq2 ).Fixing andlettingpvary,wendthatthevolumeenclosedbythisconeisalinearfunctionofp,andAp:=rpVolD=A=3=1 6H xdx.WealsonotethatanysuchconeDisintrinsicallyatexceptattheconepointp,andthat2�Kpistheconeangleatp.4.3.MeancurvatureThemeancurvatureofadiscretesurfaceMissupportedalongtheedges.Ifeisanedge,andeDStar(e)=T1[T2,thenHe:=ZZDHdA=I@Dds=e1�e2=J1e�J2e:HereiisthenormalvectortothetriangleTi,andJiisrotationby90intheplaneofthattriangle.NotethatjHej=2jejsine 2whereeistheexteriordihedralanglealongtheedge,denedbycose=12.NononplanardiscretesurfacehasHe=0alongeveryedge.Butthisdiscretemeancurvaturecancanceloutaroundthevertices.Weset2Hp:=Xe3pHe=ZZStar(p)HdA=ILink(p)ds:Theareaofthediscretesurfaceisafunctionofthevertexpositions;ifwevaryonlyonevertexp,wendthatrpArea(M)=�Hp. 8JohnM.Sullivangeneral,ofcourse,thevectorsHpandAparenotevenparallel:theygivetwocompetingnotionsofanormalvectoratp.Still,hp:=jrpAreaj jrpVolj=jHpj jApj=jRRStarpHdAj jRRStarpdAjgivesabetternotionofmeancurvaturenearpthan,say,thesmallerquantity3jHpj=Area(Star(p))=jRRHdAj=RR1dA.Forthisreason,agooddiscretizationoftheWillmoreelasticenergyRRH2dAisgivenbyPph2p1 3Area(Star(p)).4.6.RelationtodiscreteharmonicmapsDiscreteminimalsurfacesminimizearea,butalsohaveotherpropertiessimilartothoseofsmoothminimalsurfaces.Forinstance,inaconformalparameterization,theircoor-dinatefunctionsareharmonic.Wedon'tknowwheningeneraladiscretemapshouldbeconsideredconformal,buttheidentitymapiscertainlyconformal.WehavethatMisdiscreteminimalifandonlyifId:M!R3isdiscreteharmonic.HereaPLmapf:M!NiscalleddiscreteharmonicifitisacriticalpointfortheDirichletenergyE(f):=PTjrfTj2AreaM(T):WendthatE(f)�AreaNisameasureofnoncon-formality.Fortheidentitymap,E(IdM)=Area(M)andrpE(IdM)=rpArea(M)conrmingthatMisminimalifandonlyifIdMisharmonic.5.VectorBundlesonPolyhedralManifoldsHerewegiveageneraldenitionofforvectorbundlesandconnectionsonpolyhedralmanifolds;thisleadstoanotherinterpretationoftheGaußcurvatureforapolyhedralsurface.Here,apolyhedraln-manifoldPnwillmeanaCW-complexwhichishomeomor-phictoann-dimensionalmanifold,andwhichisregularandsatisestheintersectioncondition(see[Zie06]).Thatis,eachfacet(n-cell)isembeddedinPnwithnoidentica-tionsonitsboundary,andtheintersectionofanytwofaces(ofanydimension)isasingleface(ifnonempty).Denition.AdiscreterankkvectorbundleVkoverPnconsistsofavectorspaceVf=RkforeachfacetfofP.AconnectiononVkisachoiceofisomorphism rbetweenVfandVf0foreachridge(or(n�1)–cell)ofP.Here,theridgeristheintersectionofthetwofacetsfandf0.WearemostinterestedinthecasewherethevectorspacesVfhaveinnerproducts,andtheisomorphisms rareorthogonal.Considerthecasen=1,wherePisapolygonalcurve.Onanarc(oropencurve)anyvectorbundleistrivial.Onaloop(orclosedcurve),arankkvectorbundleisdeter-mined(uptoisomorphism)simplybyitsholonomyaroundtheloop,anautomorphism :Rk!Rk. 10JohnM.Sullivan[Zie06]G¨unterM.Ziegler,Polyhedralsurfacesofhighgenus,DiscreteDifferentialGeometry(Bobenko,Schr¨oder,Sullivan,andZiegler,eds.),Birkh¨auser,2006,thisvolume.JohnM.Sullivane-mail:sullivan@math.tu-berlin.deInstitutf¨urMathematik,MA3–2TechnischeUniversit¨atBerlin,DE–10623Berlin