 Journal of Applied Mathematics and Physics, 2013, 1, 55-59 http://dx.doi.org/10.4236/jamp.2013.13009 Published Online August 2013 (http://www.scirp.org/journal/jamp) Some Results on the Differential Geometry of Spacelike Curves in De-Sitter Space Tunahan Turhan1*, Nihat Ayyildiz2 1Seydişehir Vocational School, Necmettin Erbakan University, Konya, Turkey 2Department of Mathematics, Süleyman Demirel University, Isparta, Turkey Email: *tturhan07@gmail.com, *tturhan@konya.edu.tr, nihatayyildiz@sdu.edu.tr, ayyildiz67@gmail.com Received June 14, 2013; revised July 15, 2013; accepted September 1, 2013 Copyright © 2013 Tunahan Turhan, Nihat Ayyildiz. This is an open access article distributed under the Creative Commons Attribu-tion License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT The differential geometry of curves on a hypersphere in the Euclidean space reflects instantaneous properties of sphere- cal 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. Keywords: De-Sitter Space; Frenet Equations; Frenet Reference Frame; Geodesic Curvature and Torsion; Local Canonical Form 1. Introduction Let 41234,,,,1 4ixxxxxx i  be a 4-di- mensional vector space. The Lorentzian space 41,, 4 is the 4-dimensional vector space endowed with the pseudo scalar product 411223 344,xyxyxyxyxy  where 1234,,, ,xxxxxx,,, ,yyyyy4141,1234 . The norm of a vector is defined by ,.xxxLet denote the 3-dimensional unitary de-Sitter space, that is, is the hyperquadric, [2,3], 3134113411,1xRxx . Given 3-vector 1234,,, ,xxxxx,41.1234,,, ,yyyyy 1234 in Then we can define the wedge product ,,,zzzzzxyz as follows 123 4123 4123 4123 4eee exxxxxyz yyy yzzz z where is the canonical basis of . 1234,,,eeee41,A spherical displacement can be specified by a unit vector ,,xyzuuuu. along the axis of the rotation and a rotation angle  The Euler parameters of the rotation defined in terms of and u can be used to prescribe a mapping of this rotation to a point in a higher dimen- sional space [4-6]. The vector function ,1234,,,XuXXXX is given by 1234cosh ,sinh,22sinh,sinh .22xyzXXuXuX u Let ,,xyzuuuu denote a timelike rotation axis. So, we get 2222123422222coshsinh 1.22xyzXXXXuuu  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 X41.31.,Xu . For this, we will examine the differential geometry of curves on So, we will in- troduce a Frenet frame for the curve and define the geo- desic 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 31.Xt. For this aim, *Corresponding author. Copyright © 2013 SciRes. JAMP T. TURHAN, N. AYYILDIZ 56 we will use the exterior algebra of multivectors. 2. The Frenet Reference Frame Let us consider a general parametrized spacelike curves on denoted by 31Xt. We will focus on the geomet- ric properties of Xt. For this, we define arclenght parameter s as 0dd.dtXsttt (2.1) The integrand of Equation (2.1) is the magnitude of the velocity of the point as it moves along the curve XXt. If d0dXt then the function st can be in- verted to obtain ts which allows the reparameteriza- tion Xts Xs. The magnitude of ddXs is ddd 1.dddXts tssst Now, we will use the unit speed form Xs 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 . tThe Frenet frame of Xs is the set of unit vectors, E, , and TNB defined in the following way. The first vector, E is directed along the radius of the hy-perquadric and is given by .EXs Note that 1Xs  and E is a spacelike vector. The second vector, , is tangent to TXs and to It is obtained by 31.ddXTs and is a spacelike vector. So, the curve XsT in is a spacelike curve. On the other hand, since is a unit 31spacelike vector, its derivative ddTswill be normal to . So, TddTs will have a component along E given by d,dTEs which we compute by expanding the identity d,dTEs0 and we get d,,dTETTs The remaining component of ddTs orthogonal to both E and T is chosen as the direction of the unit timelike vector , so we have Ndd.ddTEsNTEs Here, we define the function d,dgTEs which measures the bend of Xs out of the ET plane, to be the geodesic curvature of .Xs The remaining vector B of the Frenet Frame is ob- tained by commuting the component of ddNs which is not along either E or and choose the direction of TB along this component such that the frame taken in the order , , TNB, E has positive orientation. The fact that the component of ddNs in the direction E is zero is obtained by expanding the identity d,0,dd,,dNEsNENTs0. On the other hand, by expanding d,dNTs0 we obtain dd,, ,dd,,gggNTTN NNssNN NE.E  Therefore we find that the component of ddTs along is T.g Finally we see that B is given by dd.ddggNTsBNTs The function ddgNTsg is defined as the geo- desic torsion of .Xs The coefficient  is chosen as either 1 or 1 to ensure that the determinant of the matrix ,,TN ,EB is 1, 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 rea- sons, the primary one being that it is convenient to visu- 1. Copyright © 2013 SciRes. JAMP T. TURHAN, N. AYYILDIZ 57alize the 3-dimensional surface of the hyperquadric lo- cally as the 3-dimensional Lorentzian space of its tangent hyperplane. The vectors T, and NB lie in this space and are analogous to the tangent, normal and bi- normal 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 Xs be a spacelike curve in de-Sitter space Then the Frenet equations are 31.ddddddd.ggNdTsNTBsBsETsggNE These equations may be viewed as a set of 16 linear first-order differential equations in the components of , , TNB and E which, when the coefficients gs and gs are specified functions of s, can be solved to determine the curve EXs in Thus the geo-desic curvature and torsion functions, 31.gs and gs, of Xs define it completely. 3. The Local Canonical Form The local properties of a hyperquadrical curve Xs in the vicinity of a reference point 0ss can be obtained by computing the series expansion of Xs in the Fre-net frame of the reference point 0.Xs This form of Xs,e is termed the local canonical form by Do Carmo . We choose the coordinate directions of the 4-dimen- sional Lorentzian space containing denoted by 1 2 3 4 where i has a in the i-th coordinate position and zeros elsewhere, so that they align with the Frenet frame 4131,1,e,e,e e0,Ts Ns0,0Bs and 0Es of the reference point 0.Xs Computing the derivatives of Xs to the third order we have 22323,d,ddd ,1TNsddddggggXs EXTsXT NEssXBs dd. These expressions lead to the Taylor series expansion of Xs in the vicinity of the reference position 0ss. For convenience, we denote the reference position as 00s and obtain   00232232211 100 001! 2!3!0110 1001 11000 26d00 d01101001000 200gggggggsXX sXsXsss ssss                  X00136gggs (3.1) where 0g and 0g are the values of the curvature and torsion of Xs and 1g is the value of ddgs, all evaluated at the reference position Equation (3.1) allows a description of the shape of 0.sXs to various of approximation, for example, to the zeroth order Xs is simply the point 0X, to the first order it is ap- proximated by the tangent vector . We see that 0Tg defines the shape of Xs to the second order which defines the amount that it bends away from the ET plane. The parameter 0g defines the amount that Xs bends out of the subspace. To the second order, ETNXs is approximated by its osculating circle which has the radius  given by 12211.1g The function 1221g  is called the total cur-vature of .Xs The plane of this circle osculating plane, is defined by the tangent vector and the unit vector G Tdd 1.ddgTsGNTsE (3.2) The rotation of the osculating plane about is given by TddGs where ddgggGTGsB  (3.3) and 21ggGN .E (3.4) If 10g and 00g then from (3.3) we see that Copyright © 2013 SciRes. JAMP T. TURHAN, N. AYYILDIZ 58 this plane remains instantaneously fixed, i.e. d.dGTs Furthermore, since d0dgs implies d0ds and therefore that the total curvature is constant. A general curve Xs is approximated to the third order by an osculating sphere To determine the center c of this pseudo-sphere we first note that first, second and third derivatives of 21.Xs lie in the subspace spanned by the three orthogonal unit vectors T, and , i.e. we have G*G22ddddgXTsXNE Gs and 32*3dd ddddXGTGsssG where *1gggggggggGBGGGB B and 222 .ggggggGB   Here is a spacelike vector if if not, respectively. *G222 0,gg Assume that is a timelike vector. So, the radius vector, , of the osculating sphere must have the form *GsRX c*123RXsckTkGkG where 123 are constants. These constants are de- termined by the requirement that ,,kkkR have constant magnitude to the third order. Differentiating ,const.RR we obtain 223232d,0,dddd,,dddddd,3,dddXXcsXXXXc sssXXXc sss  0,0.X So, from these equations we conclude 1230,1,.kkk Hence, we have *211.gggXs cGGGG  B For those curves with 0g this relation simplifies using (3.1) and (3.3) 22211 .ggg2XscN E     (3.5) So we have the following proposition. Proposition 3.1. Let Xs be a spacelike curve with the geodesic curvature g and the total curvature  in de-Sitter space Then the radius vector 31.RXsc of the osculating sphere is as in Equation (3.5). 214. Arbitrary Parameterization We now derive the local canonical form of Xt with respect to the arbitrary parameter . To do this we use the Frenet equations and the fact that tdd ddd..dddddXt XtXtssvtstst Computing the derivatives of Xt to the third order, we obtain 222232323233,d,ddd,dddd1ddddd33dd dggggggXtEXt vTtXt vTvNvEttXt vvTttvvvvNvBvts  .Et(4.1) Thus the local canonical form for Xt at a refer-ence point 0t becomes, to third order, Copyright © 2013 SciRes. JAMP T. TURHAN, N. AYYILDIZ Copyright © 2013 SciRes. JAMP 59 000010232232232223231d 1d1d00 001! d2!3!ddd3d10 dd101 dd.00 26000 dd3ddggggggXX X33XtXt tttttvvtvvvvttttvvvvvts         6. Conclusion tThis work develops the differential geometry of space- like curves on de-Sitter space in four-dimensional Lor- entzian 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 31. REFERENCES  B. O’Neill, “Semi-Riemann Geometry: With Applictions to Relativity,” Academic Press, New York, 1983, 469 p. 5. Formula for Geodesic Curvature and Torsion  T. Fusho and S. Izumiya, “Lightlike Surfaces of Space- like Curves in de Sitter 3-Space,” Journal of Geometry, Vol. 88, 2008, pp. 19-29. http://dx.doi.org/10.1007/s00022-007-1944-5 We now compute the vector 22ddddXXXtt which in view of Equation (4.1) becomes  M. Kasedou, “Singularities of Lightcone Gauss Images of Spacelike Hypersurfaces in de Sitter Space,” Journal of Geometry, Vol. 94, 2009, pp. 107-121. http://dx.doi.org/10.1007/s00022-009-0001-y 232dd .ddgXXXvTNEtt  (5.1)  J. M. McCarthy, “The Differential Geometry of Curves in an Image Space of Spherical Kinematics,” Mechanism and Machine Theory, Vol. 22, No. 3, 1987, pp. 205-211. http://dx.doi.org/10.1016/0094-114X(87)90003-6 Computing the scalar product of (5.1) with itself, we obtain the following equation for the geodesic curvature g:  J. M. McCarthy and B. Ravani, “Differential Kinematics of Spherical and Spatial Motions Using Kinematic Map- ping,” Journal of Applied Mechanics, Vol. 53, No. 1, 1986, pp. 15-22. http://dx.doi.org/10.1115/1.3171705 222223dd dd,dddd.dd,ddgXX XXXXttttXXtt   B. Ravani and B. Roth, “Mappings of Spatial Kinemat- ics,” Journal of Mechanisms, Transmissions and Automa- tion in Design, Vol. 106, No. 3, 1984, pp. 341-347. http://dx.doi.org/10.1115/1.3267417 And a formula for geodesic torsion g can be given as 2322dd ddet,,,.dddgg3XXXEttt   M. P. Do Carmo, “Differential Geometry of Curves and Surfaces,” Prentice-Hall, Englewood Cliffs, 1976, 503 p. This relation can be also seen in .