Mathematica MoravicaVol. 19-1 (2015), 8793Spinor Darboux Equations ofCurves in Euclidean 3-SpaceIlim Kii and Murat TosunAbstract. In this paper, the spinor formulation of Darboux frame onan oriented surface is given. Also, the relation between spinor formula-tion of Frenet frame and Darboux frame is obtained.1. IntroductionSpinorsingeneral werediscoveredbyElieCartanin1913[2]. Later,spinors were adopted by quantum mechanics in order to study the proper-ties of the intrinsic angular momentum of the electron and other fermions.Today, spinors enjoy a wide range of physics applications. In mathematics,particularly in dierential geometry and global analysis, spinors have sincefound broad applications to algebraic and dierential topology,symplecticgeometry, gauge theory, complex algebraic geometry, index theory [13, 11].Inthedierential geometryofsurfacestheDarbouxframeisanaturelmoving frame constructed on a surface. It is the analog of the Frenet- Serretframe as applied to surface geometry. A Darboux frame exists at any non-umbilic point of a surface embedded in Euclidean space. The construction ofDarboux frames on the oriented surface rst considers frames moving alongacurveinthesurface, andthenspecializeswhenthecurvesmoveinthedirection of the principal curvatures [10].In[2, 4], thetriadsofmutuallyorthogonal unitvectorswereexpressedin terms of a single vector with two complex components, which is called aspinor. In the light of the existing studies, reducing the Frenet equations toa single spinor equation,equivalent to the three usual vector equations,isa consequence of the relationship between spinors and orthogonal triads ofvectors. The aim of this paper is to show that the Darboux equations can beexpressed with a single equation for a vector with two complex components.2010 Mathematics Subject Classication. Primary: 53A04; Secondary: 15A66.Key words and phrases. Frenet equations; Darboux equations; spinors.c 2015MathematicaMoravica8788 SpinorDarbouxEquationsofCurvesinEuclidean3-Space2. PreliminariesThe Euclidean 3-space provided with standard at metric is given by, = dx21 +dx22 +dx23,where (x1, x2, x3) is a rectangular coordinate system of E3. Recall that, thenormofanarbitraryvectorx E3isgivenby |x|=

x, x. Letbea curve in Euclidean 3- space. The curve is called a unit speed curve ifvelocity vector

of satises |

| = 1.Let us denote by T(s),N(s) and B(s) the unit tangent vector, unit normalvector and unit binormal vector of respectively. The Frenet Trihedron isthe collection ofT(s),N(s) andB(s). Thus, the Frenet formulas are asd Tds= (s)N(s)d Nds= (s)T(s) +(s)B(s)d Bd s= (s)N(s).Here,the curvature is dened to be(s)= |T

(s)| and the torsion is thefunctionsuch thatB

(s) = (s)N(s) [8].The group of rotation about the origin denoted bySO(3) inR3is homo-morphic to the group of unitary complex 22 matrices with unit determinantdenoted by SU(2). Thus, there exits a two-to-one homomorphism of SU(2)ontoSO(3). Whereas the elements ofSO(3) act the vectors with three realcomponents, the elements ofSU(2) act on vectors with two complex com-ponents which are called spinors, [1, 5]. In this case, we can dene a spinoras(1) =

12

by means of three vectorsa, b, c R3such that(2) a +ib = t, c = t.Here, the superscript t denotes transposition and = (1, 2, 3) is a vectorwhose cartesian components are the complex symmetric22 matrices(3) 1=

1 00 1

, 2=

i 00 i

, 3=

0 11 0

.Inadditiontothis,isthemate(orconjugate)of andiscomplexconjugation of [16]. Therefore,(4)

0 11 0

=

21

.IlimKiiandMuratTosun 89In that case, the vectorsa, b andc are explicitly given bya +ib = t= (21 22, i(21 +22), 212),c = t= (1 2 +12, i(1 2 12), [1[2 [2[2). (5)Sincethevector a + ib C3isanisotropicvector, bymeansof aneasycomputation one can nd thata, b andc are mutually orthogonal [2, 4, 6].Also, [a[ = [b[ = [c[ =tand=det(a, b, c) >0. Onthecontrary,if the vectorsa, b andc are mutually orthogonal vectors of samemagnitudes such that (det(a, b, c) > 0), then there is a spinor, dened up tosign such that the equation (2) holds. Under the conditions stated above,for two arbitrary spinors and, there exist the following equalitiest= t ,

a +b= a +b, (6)and= ,whereaandbarecomplexnumbers. Thecorrespondencebetweenthespinorsandtheorthogonal basesgivenby(2)istwotoone, becausethespinors and correspond to the same ordered orthonormal bases a, b, c,with [a[ = [b[ = [c[, and< a b, c >. In addition to that, the ordered triads{a,b,c}, {b,c,a} and {c,a,b} correspond to dierent spinors. Since the ma-trices(given by (3)) are symmetric, any pairs of spinors andsatisfyt= t. The set, is linearly independent for the spinor ,= 0 [4].3. SpinorDarbouxEquationsIn this section we investigate the spinor equation of the Darboux equationsfor a curve on the oriented surface in Euclidean 3-spaceE3. Let Mbe anoriented surface in Euclidean 3-space and let us consider a unit speed regularcurve (s) on the surface M. Since the curve (s) is also in space, there existsFrenetframe T,N,BateachpointofthecurvewhereTisunittangentvector, Nis principal normal vector and Bis binormal vector, respectively.The Frenet equations of curve(s) is given byd Td s= Nd Nd s= T+B (7)d Bd s= Nwhere andare curvature and torsion of the curve(s), respectively [2].According to the result concerned with the spinor (given by section 2) there90 SpinorDarbouxEquationsofCurvesinEuclidean3-Spaceexists a spinor such that(8) N+iB= t, T= twitht=1. Thus, thespinorrepresentsthetriad{N,B,T}andthevariationsofthistriadalongthecurve(s)mustcorrespondtosomeex-pression ford d s. That is, the Frenet equations are equivalent to the singlespinor equation(9)d d s=12(i +)where anddenote the torsion and curvature of the curve, respectively.The equation (9) is called spinor Frenet equation, [4].Since the curve(s) lies on the surface M, there exists another frame ofthe curve(s) which is called Darboux frame and denoted by {T,g,n}. Inthis frame, Tis the unit tangent of the curve, nis the unit normal of thesurface Mand gis a unit vector given by g= nT. Since the unit tangentTis common in both Frenet frame and Darboux frame, the vectors N,B,g, and nlie on the same plane. So that, the relations between these framescan be given as follows(10)Tgn =1 0 00 cos sin 0 sin cos TNBwhere is the angle between the vectors gand N. The derivative formulaeof the Darboux frame is(11)d Td sd gd sd nd s =0 gng0 gng0Tgnwhereg is the geodesic curvature,kn is the normal curvature andg is thegeodesic torsion of(s) [10].Considering the equations (1), (2) and (8) there exists a spinor , denedup to sign, such that(12) g +in = t, T= tandt = 1.That is, the spinor represents the triad {g,n,T} of the curve(s) on thesurface M. The variations of the triad {g,n,T} along curve must correspondtosomeexpressionfordds. Since , isabasisforthetwocomponentspinors( ,= 0), there are two functions f and h, such that(13)d d s= f +hIlimKiiandMuratTosun 91where the functions f and g are possibly complex-valued functions. Dier-entiating the rst equation in (12) and using the equation (13) we haved gd s+id nd s=

d d s

t +t

d d s

=2f(t) + 2h(t).Substitutingtheequations(11)and(12)intothelastequationandaftersimplifying, one nds(14) ig(g +in) + (g in)T= 2f(g +in) 2hT.From the last equation we get(15) f= ig2 , h =in +g2Thus, we have proved the following theorem.Theorem3.1. Letthetwo-componentspinorrepresentsthetriad{g,n,T}ofa curve parametrized by arc length on the surface M. Then, the Darboux equationsare equivalent to the single spinor equation(16)d d s=

ig2

+

in +g2

where nandgarenormal andgeodesiccurvature, respectivelyandgisthegeodesic torsion of(s).The equation (16) is called spinor Darboux equation. Now, we investigatethe relation between the spinors and represent the triad {N,B,T} (Frenetframe) and {g,n,T} (Darboux Frame), respectively.Considering the equation (10) we get(17) N+iB= (g +in)(cos +i sin )Thus, from the equations (8),(12) and (17) we can give the following theorem.Theorem 3.2.Let the spinors and represent the triads {N,B,T} and {g,n,T}ofacurveparametrizedwitharclength, respectively. Then, therelationbetweenthe spinor formulation of Frenet frame and Darboux frame is as followst=ei(t)T=T4. ConclusionsIn this study, we have established the spinor Darboux equations of curveson an oriented surface inE3and then we have given the relation betweenspinorformulationsof FrenetframeandDarbouxframe. Also, wehaveshown that the Darboux equations are equivalent to a single spinor equation,which is a consequence of the relationship between spinors and orthogonaltriads of vectors, and of using complex quantities.92 SpinorDarbouxEquationsofCurvesinEuclidean3-SpaceThere are so many applications of Darboux frame in dierential geometry.So far, dierent curves such as Bertrand curves[7], biharmonic D-helices [15],Smarandachecurves[12] havebeenstudiedaccordingtoDarbouxframe.Furthermore, other studies like inextensible ows of these curves [14], tubes[3], dual Darboux frames of ruled surfaces have been given [9].Thesekindsofapplicationscanbeincludedinfutureinvestigationsbymeansof spinorDarbouxequationsof curvesonsurfaces. Moreover, thespinor formulations of curves can be given not only in Euclidean 3-space butalsoinotherthreedimensional spaceslikeMinkowski 3-spaceorGalilean3-space, etc.References[1] D.H. Sattinger andO.L. Weaver, Lie Groups andAlgebras withApplications toPhysics, Geometry and Mechanics, Springer-Verlag, New York, 1986.[2] E. Cartan, The theory of spinors, Hermann, Paris, 1966. (Dover, New York, reprinted1981)[3] F. Doan, Tubes with Darboux Frame,Int. J. Contemp. Math. Sciences, Vol. 7(2012),751-758.[4] G.F.T. Del Castillo and G.S. Barreles, Spinor formulation of the dierential geometryof curves,Revista Colombiana de Matematicas, Vol. 38(2004), 27-34.[5] H. Goldstein, Classical mechanics, 2nd ed., Addison-Wesley, Reading, Mass., 1980.[6] H.B. Lawson and M.L. Michelsohn, Spin Geometry, Princeton University, 1989.[7] M. Babaarslan, Y.A. Tandoan and Y. Yayl, A Note on Bertrand Curves and Con-stant Slope Surfaces According To Darboux Frame,J. Adv. Math. Stud., Vol. 5(2012),87-96.[8] M.P.DoCarmo, Dierential GeometryofCurvesandSurfaces, PrenticeHall, En-glewood Clis, NJ, 1976.[9] M. nder and H.H. Uurlu, Dual Darboux Frame of a Spacelike Ruled Sur-face and Darboux Approach to MannheimOsets of Spacelike Ruled Surfaces,arxiv:1108.6076,(2011) [math.DG].[10] M. Spivak, A Comprehensive introduction to dierential geometry, Publish or Perish,Boston, 1970.[11] N.J. Hitchin, Harmonics spinors, Advanced in Mathematics, Vol. 14(1974), 1-55.[12] . Bekta and S. Yce, Special Smarandache Curves According to Frame inE3,Romanian Journal of Mathematics and Computer Science, Vol. 3(2013), 48-59.[13] P.B. Gilkey, InvarianceTheory, theHeat Equation, andtheAtiyah-Singer IndexTheorem, Publish or Perish, 1984.[14] S. Ba and T. Krpnar, nextensible Flows of Spacelike Curves on Spacelike Surfacesaccordingto Darboux Frame inM31, Bol. Soc. Paran. Mat., Vol. 31(2013), 9-17.IlimKiiandMuratTosun 93[15] T. KrpnarandE. Turhan, Oncharacterizationof timelikebiharmonicD-helicesaccordingtoDarbouxframeonnon-degeneratetimelikesurfacesintheLorentzianHeisenberggroupH, Annalsof FuzzyMathematicsandInformatics, Vol. 4(2012),393-400.[16] W.T. Payne, Elementary spinor theory, Am. J. of Physics, Vol. 20(1952), 253-262.IlimKiiFacultyofArtandSciencesDepartmentofMathematicsKocaeliUniversityTR41380KocaeliTurkeyE-mail address: ilim.ayvaz@kocaeli.edu.trMuratTosunFacultyofArtandSciencesDepartmentofMathematicsSakaryaUniversityTR54187SakaryaTurkeyE-mail address: tosun@sakarya.edu.tr