Spinor Darboux Equations of Curves in Euclidean 3-Space

In this paper, the spinor formulation of Darboux frame on an oriented surface is given. Also, the relation between the spinor formulation of Frenet frame and Darboux frame are obtained.


Introduction
Spinors in general were discovered by Elie Cartan in 1913 [3]. Later, spinors were adopted by quantum mechanics in order to study the properties of the intrinsic angular momentum of the electron and other fermions. Today, spinors enjoy a wide range of physics applications. In mathematics particularly in differential geometry and global analysis, spinors have since found broad applications to algebraic and differential topology, symplectic geometry, gauge theory, complex algebraic geometry, index theory [5,7].
In the differential geometry of surfaces a Darboux frame is a naturel moving frame constructed on a surface. It is the analog of the Frenet-Serret frame as applied to surface geometry. A Darboux frame exists at any non-umbilic point of a surface embedded in Euclidean space. The construction of Darboux frames on the oriented surface first considers frames moving along a curve in the surface, and then specializes when the curves move in the direction of the principal curvatures [9].
In [3,4], the triads of mutually orthogonal unit vectors were expressed in terms of a single vector with two complex components, called a spinor. In the light of the existing studies, the Frenet equations reduce to a single spinor equation, equivalent to the three usual vector equations, is a consequence of the relationship between spinors and orthogonal triads of vectors. The aim of this paper is to show that the Darboux equations can be expressed a single equation for a vector with two complex components.

Preliminaries
The Euclidean 3-space provided with standard flat metric given by where (x1, x2, x3) is a rectangular coordinate system of E 3 . Recall that, the norm of an arbitrary vector x ∈ E 3 is given by x = x, x . Let α be a curve in Euclidean 3-space. The curve α is called a unit speed curve if velocity vector α ′ of α satisfies α ′ = 1.
Let us denote by T (s),N (s) and B(s) the unit tangent vector, unit normal vector and unit binormal vector of α respectively. The Frenet Trihedron is the collection of T (s),N (s) and B(s). Thus, the Frenet formulas are as Here, the curvature is defined to be κ(s) = T ′ (s) and the torsion is the function τ such that B ′ (s) = −τ (s)N (s) [2].
The group of rotation about the origin denoted by SO (3) in R 3 is homomorphic to the group of unitary complex 2 × 2 matrices with unit determinant denoted by SO(2). Thus, there exits a two-to-one homomorphism of SO(2) onto SO (3). Whereas the elements of SO(3) act the vectors with three real components, the elements of SO(2) act on vectors with two complex components which are called spinors [1,6]. In this case, we can define a spinor Here, the superscript t denotes transposition and σ = (σ1, σ2, σ3) is a vector whose cartesian components are the complex symmetric 2 × 2 matrices In addition to this,ψ is the mate (or conjugate) of ψ andψ is complex conjugation of ψ [10]. Therefore,ψ In that case, the vectors a, b and c are explicitly given by (2.5) Since the vector a + ib ∈ C 3 is an isotropic vector, by means of an easy computation one find that a, b and c are mutually orthogonal [3,4,8]. Also, |a| = |b| = |c| =ψ t ψ and < a ∧ b, c >=det(a, b, c) > 0. On the contrary, for the vectors a, b and c mutually orthogonal vectors of same magnitudes (det(a, b, c) > 0) there is a spinor defined up to sign such that the equation (2.2) holds. Under the conditions state above, for two arbitrary spinors φ and ψ, there exist following equalities aφ + bψ =āφ +bψ, (2.6) andψ = −ψ.
where a and b are complex numbers. The correspondence between spinors and orthogonal bases given by (2.2) is two to one; the spinors ψ and −ψ 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,a,c} and {c,a,b} correspond to different spinors. Since the matrices σ (given by (2.3)) are symmetric, any pair of spinors φ and ψ satisfying φ t σψ = ψ t σφ. The set ψ,ψ is linearly independent for the spinor ψ = 0 [4].

Spinor Darboux Equations
In this section we investigate the spinor equation of the Darboux equations for a curve on the oriented surface in Euclidean three 3-space E 3 . Let M be an oriented surface in Euclidean 3-space and let consider a curve α(s) on the surface M. Since the curve α(s) is also in space, there exists Frenet frame T,N,B at each points of the curve where T is unit tangent vector, N is principal normal vector and B is binormal vector, respectively. The Frenet equations of curve α(s) is given by where κ and τ are curvature and torsion of the curve α(s), respectively [3]. Unless otherwise stated we assume that the curve α(s) is a regular curve with arc length parameter. According to the result concerned with the spinor (given by section 2) there exists a spinor ψ such that withψ t ψ = 1. Thus, the spinor ψ represent the triad {N,B,T} and the variations of this triad along the curve α(s) must correspond to some expression for dψ ds . That is, the Frenet equations are equivalent to the single spinor equation where κ and τ denote the torsion and curvature of the curve, respectively. The equation (3.3) is called spinor Frenet equation, [4].
Since the curve α(s) lies on the surface M, there exists another frame of the curve α(s) which is called Darboux frame and denoted by {T,g,n}. In this frame, T is the unit tangent of the curve, n is the unit normal of the surface M and g is a unit vector given by g = n ∧ T . Since the unit tangent T is common in both Frenet frame and Darboux frame, the vectors N, B, g, and n lie on the same plane. So that, the relations between these frames can be given as follows  where θ is the angle between the vectors g and N. The derivative formulae of the Darboux frame is  where κg is the geodesic curvature, kn is the normal curvature and τg is the geodesic torsion of α(s) [9]. Considering the equations (2.1), (2.2) and (3.2) there exists a spinor φ, defined up to sign, such that That is, the spinor φ represent the triad {g,n,T } of the curve α(s) on the surface M.
The variations of the triad {g,n,T } along curve must correspond to some expression for dφ ds . Since {φ,φ} is a basis for the two component spinors (φ = 0), there are two functions f and g, such that dφ where the functions f and g are possibly complex-valued functions. Differentiating the first equation in (3.6) and using the equation (3.7) we have Substituting the equations (3.5) and (3.6) into the last equation and after simplifying, From the last equation we get Thus, we have proved the following theorem.
where κn and κg are normal and geodesic curvature, respectively and τg is the geodesic torsion of α(s).  T =T