Some Results on the Differential Geometry of Spacelike Curves in De-Sitter Space

The differential geometry of curves on a hypersphere in the Euclidean space reflects instantaneous properties of spherecal motion. In this work, we give some results for differential geometry of spacelike curves in 3-dimensional de-Sitter space. Also, we study the Frenet reference frame, the Frenet equations, and the geodesic curvature and torsion functions to analyze and characterize the shape of the curves in 3-dimensional de-Sitter space.


Introduction
[1].The norm of a vector is defined by , .
x x x  Let denote the 3-dimensional unitary de-Sitter space, that is, is the hyperquadric, [2,3], , , , , , , , , y y y y y   can be used to prescribe a mapping of this rotation to a point in a higher dimensional space [4][5][6].The vector function This means that the point lies on the hyperquadric of radius 1 in Let us denote this hyperquadric Our aim is to give an interpretation of the image of the mapping  For this, we will examine the differential geometry of curves on So, we will introduce a Frenet frame for the curve and define the geodesic curvature and geodesic torsion functions which characterize the shape of the curve.Also we will give explicit formulas for the geodesic curvature and torsion functions of the parameterized curve 3 1 .   X t .For this aim, we will use the exterior algebra of multivectors.

The Frenet Reference Frame
Let us consider a general parametrized spacelike curves on denoted by . We will focus on the geometric properties of X t .For this, we define arclenght parameter s as The integrand of Equation (2.1) is the magnitude of the velocity of the point as it moves along the curve Now, we will use the unit speed form

 
X s to define the Frenet frame and the Frenet equations of the curve.And so, we will give interpretation of these results in terms of the general parameter .
t The Frenet frame of   X s is the set of unit vectors, E , , and T N B defined in the following way.The first vector, E is directed along the radius of the hyperquadric and is given by On the other hand, by expanding Therefore we find that the component of d d , that is so that the Frenet frame has positive orientation.The vector E has been choosen as the last member of the frame for several reasons, the primary one being that it is convenient to visu-1 .
alize the 3-dimensional surface of the hyperquadric locally as the 3-dimensional Lorentzian space of its tangent hyperplane.The vectors T , and N B lie in this space and are analogous to the tangent, normal and binormal vectors of a space curve in three dimensions.In this way the geodesic curvature and torsion functions g  and g  are seen to be analogous to the curvature and torsion of a space curve.
Then we have the following proposition.Proposition 2.1.Let

 
X s be a spacelike curve in de-Sitter space Then the Frenet equations are These equations may be viewed as a set of 16 linear first-order differential equations in the components of , ,

The Local Canonical Form
The local properties of a hyperquadrical curve where i has a in the i-th coordinate position and zeros elsewhere, so that they align with the Frenet frame Computing the derivatives of   X s to the third order we have These expressions lead to the Taylor series expansion of   X s in the vicinity of the reference position 0 s s  .
For convenience, we denote the reference position as 0 0 s  and obtain

 
X s is approximated by its osculating circle which has the radius  given by   X s The plane of this circle osculating plane, is defined by the tangent vector and the unit vector The rotation of the osculating plane about is given by Assume that is a timelike vector.So, the radius vector, , of the osculating sphere must have the form where 1 2 3 are constants.These constants are determined by the requirement that For those curves with 0 g   this relation simplifies using (3.1) and (3.3) So we have the following proposition.Proposition 3.1.Let

 
X s be a spacelike curve with the geodesic curvature g  and the total curvature  in de-Sitter space Then the radius vector of the osculating sphere is as in Equation (3.5).

Arbitrary Parameterization
We now derive the local canonical form of   X t with respect to the arbitrary parameter .To do this we use the Frenet equations and the fact that Computing the derivatives of

 
X t to the third order, we obtain Thus the local canonical form for   X t at a refer- ence point 0 t  becomes, to third order, This work develops the differential geometry of spacelike curves on de-Sitter space in four-dimensional Lorentzian space.The motivation for this work is the fact that the Euler parameters of spherical displacements can be used to map them to points on 3 1 .

Formula for Geodesic Curvature and Torsion
[2] T. This relation can be also seen in [2].
of the rotation and a rotation angle  The Euler parameters of the rotation defined in terms of and u The remaining vector B of the Frenet Frame is obtained by commuting the component of d d N s which is not along either E or and choose the direction of T B along this component such that the frame taken in the order , , T N B , E has positive orientation.The fact that the component of d d N s in the direction E is zero is obtained by expanding the identity total curvature is constant.A general curve X s is approximated to the third order by an osculating sphereTo determine the center c of this pseudo-sphere we first note that first, second and third derivatives in the subspace spanned by the three orthogonal unit vectors T , and , i.e. we have and E is a spacelike vector.