Journal of Applied Mathematics and Physics, 2013, 1, 5559 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 DeSitter 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 3dimensional deSitter 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 3dimensional deSitter space. Keywords: DeSitter Space; Frenet Equations; Frenet Reference Frame; Geodesic Curvature and Torsion; Local Canonical Form 1. Introduction Let 4 1234 ,,,,1 4 i xxxxxx i be a 4di mensional vector space. The Lorentzian space 4 1 ,, 4 is the 4dimensional vector space endowed with the pseudo scalar product 4 11223 344 , yxyxyxyxy where 1234 ,,, , xxxxx ,,, ,yyyyy 4 1 4 1, 1234 [1]. The norm of a vector is defined by ,. xxLet denote the 3dimensional unitary deSitter space, that is, is the hyperquadric, [2,3], 3 1 34 11 34 11 ,1xRxx . Given 3vector 1234 ,,, , xxxx ,4 1. 1234 ,,, ,yyyyy 1234 in Then we can define the wedge product ,,,zzzzz yz as follows 123 4 123 4 123 4 123 4 eee e xxx xyz yyy y zzz z where is the canonical basis of [2]. 1234 ,,,eeee 4 1, A spherical displacement can be specified by a unit vector ,, yz uuuu . 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 [46]. The vector function , 1234 ,,, uXX XX is given by 12 34 cosh ,sinh, 22 sinh,sinh . 22 x yz XXu uX u Let ,, yz uuuu denote a timelike rotation axis. So, we get 2222 1234 22222 coshsinh 1. 22 xyz XXXX uuu 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 X 4 1. 3 1. ,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 3 1. t. For this aim, *Corresponding author. C opyright © 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 3 1 t . We will focus on the geomet ric properties of t. For this, we define arclenght parameter as 0 dd. d tX t t 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 t. If d0 d X t then the function t can be in verted to obtain ts which allows the reparameteriza tion ts Xs. The magnitude of d d is ddd 1. ddd Xts ts sst Now, we will use the unit speed form 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 s is the set of unit vectors, , , and TN defined in the following way. The first vector, is directed along the radius of the hy perquadric and is given by .EXs Note that 1Xs and is a spacelike vector. The second vector, , is tangent to T s and to It is obtained by 3 1. d d T and is a spacelike vector. So, the curve sT in is a spacelike curve. On the other hand, since is a unit 3 1 spacelike vector, its derivative d d T will be normal to . So, T d d T will have a component along given by d, d TE s which we compute by expanding the identity d, dTE s0 and we get d,, d TETT s The remaining component of d d T orthogonal to both and T is chosen as the direction of the unit timelike vector , so we have N d d. d d TE s NTE s Here, we define the function d, d gTE s which measures the bend of s out of the T plane, to be the geodesic curvature of . s The remaining vector of the Frenet Frame is ob tained by commuting the component of d d N which is not along either or and choose the direction of T along this component such that the frame taken in the order , , TN , has positive orientation. The fact that the component of d d N in the direction is zero is obtained by expanding the identity d,0, d d,, d NE s NENT s 0. On the other hand, by expanding d, dNT s0 we obtain dd ,, , dd ,, g gg NT TN NN ss NN NE . E Therefore we find that the component of d d T along is T. Finally we see that is given by d d. d d g g NT s BNT s The function d d g NT s g is defined as the geo desic torsion of . s 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 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 57 alize the 3dimensional surface of the hyperquadric lo cally as the 3dimensional Lorentzian space of its tangent hyperplane. The vectors T, and N 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 and are seen to be analogous to the curvature and torsion of a space curve. Then we have the following proposition. Proposition 2.1. Let s be a spacelike curve in deSitter space Then the Frenet equations are 3 1. d d d d d d d. g gN d T s NTB s B s ET s gg NE These equations may be viewed as a set of 16 linear firstorder differential equations in the components of , , T N and which, when the coefficients g and g are specified functions of , can be solved to determine the curve EXs in Thus the geo desic curvature and torsion functions, 3 1. g and g , of s define it completely. 3. The Local Canonical Form The local properties of a hyperquadrical curve s in the vicinity of a reference point 0 s can be obtained by computing the series expansion of s in the Fre net frame of the reference point 0. s This form of s ,e is termed the local canonical form by Do Carmo [7]. We choose the coordinate directions of the 4dimen sional Lorentzian space containing denoted by 1 2 3 4 where i has a in the ith coordinate position and zeros elsewhere, so that they align with the Frenet frame 4 1 3 1, 1 ,e,e,e e 0,Ts Ns 0, 0 Bs and 0 Es of the reference point 0. s Computing the derivatives of s to the third order we have 2 2 3 2 3 , d, d dd , 1TN s d d d d g ggg Xs E XT s XT NE s s XB s dd . These expressions lead to the Taylor series expansion of s in the vicinity of the reference position 0 s . For convenience, we denote the reference position as 00s and obtain 0 0 23 2 23 2 2 11 1 00 00 1! 2!3! 0 1 10 1 0 01 11 0 00 26 d 00 d 0 1 10 1 0 01 0 00 2 00 g gg g g g g sXX sXsXs ss s s s s 00 1 3 6 gg g s (3.1) where 0 and 0 are the values of the curvature and torsion of s and 1 is the value of d d , all evaluated at the reference position Equation (3.1) allows a description of the shape of 0.s s to various of approximation, for example, to the zeroth order s is simply the point 0X, to the first order it is ap proximated by the tangent vector . We see that 0 T defines the shape of s to the second order which defines the amount that it bends away from the T plane. The parameter 0 defines the amount that s bends out of the subspace. To the second order, ETN s is approximated by its osculating circle which has the radius given by 12 2 11 . 1 g The function 12 21 g is called the total cur vature of . s The plane of this circle osculating plane, is defined by the tangent vector and the unit vector G T dd 1 . dd g Ts GN Ts E (3.2) The rotation of the osculating plane about is given by T d d G where d d ggg GTG s B (3.3) and 2 1 gg GN .E (3.4) If 10 g and 00 g then from (3.3) we see that Copyright © 2013 SciRes. JAMP
T. TURHAN, N. AYYILDIZ 58 this plane remains instantaneously fixed, i.e. d. d GT s Furthermore, since d0 d g s implies d0 ds and therefore that the total curvature is constant. A general curve s is approximated to the third order by an osculating sphere To determine the center c of this pseudosphere we first note that first, second and third derivatives of 2 1. s lie in the subspace spanned by the three orthogonal unit vectors T, and , i.e. we have G* G 2 2 d d d dg XT s XNE G s and 3 2* 3 dd d dd d XGTG ss s G where *1 gggggg ggg GB G G GB B and 2 22 . gggggg GB Here is a spacelike vector if if not, respectively. * G 2 22 0, gg Assume that is a timelike vector. So, the radius vector, , of the osculating sphere must have the form * G sRX c * 123 RXsckTkGkG where 123 are constants. These constants are de termined by the requirement that ,,kkk have constant magnitude to the third order. Differentiating ,const.RR we obtain 2 2 32 32 d,0, d ddd ,, dd d ddd ,3, d dd XXc s XXX Xc ss s XX Xc s ss 0, 0. X So, from these equations we conclude 1 2 3 0, 1, . k k k Hence, we have * 2 1 1. ggg Xs cGG GG B For those curves with 0 g this relation simplifies using (3.1) and (3.3) 22 2 11 . ggg 2 scN E (3.5) So we have the following proposition. Proposition 3.1. Let s be a spacelike curve with the geodesic curvature and the total curvature in deSitter space Then the radius vector 3 1. RXsc of the osculating sphere is as in Equation (3.5). 2 1 4. Arbitrary Parameterization We now derive the local canonical form of t with respect to the arbitrary parameter . To do this we use the Frenet equations and the fact that t dd d dd .. ddddd Xt XtXt ss v tstst Computing the derivatives of t to the third order, we obtain 2 22 2 3232 32 33 , d, d dd, d d dd1 dd d dd 33 dd d g g g ggg XtE Xt vT t Xt vTvNvE t t Xt vvT tt vv vvNvBv ts .E t (4.1) Thus the local canonical form for t at a refer ence point 0t becomes, to third order, Copyright © 2013 SciRes. JAMP
T. TURHAN, N. AYYILDIZ Copyright © 2013 SciRes. JAMP 59 0 00 01 0 23 2 23 2 232 2 2 3 2 3 1d 1d1d 00 00 1! d2!3! dd d 3d 10 d d1 01 d d. 00 26 0 00 d d 3dd g gg gg g XX X 3 3 tXt tt ttt v vt vv vv t t t tv vv vv ts 6. Conclusion t This work develops the differential geometry of space like curves on deSitter space in fourdimensional 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 3 1. REFERENCES [1] B. O’Neill, “SemiRiemann Geometry: With Applictions to Relativity,” Academic Press, New York, 1983, 469 p. 5. Formula for Geodesic Curvature and Torsion [2] T. Fusho and S. Izumiya, “Lightlike Surfaces of Space like Curves in de Sitter 3Space,” Journal of Geometry, Vol. 88, 2008, pp. 1929. http://dx.doi.org/10.1007/s0002200719445 We now compute the vector 2 2 dd dd XX tt which in view of Equation (4.1) becomes [3] M. Kasedou, “Singularities of Lightcone Gauss Images of Spacelike Hypersurfaces in de Sitter Space,” Journal of Geometry, Vol. 94, 2009, pp. 107121. http://dx.doi.org/10.1007/s000220090001y 2 3 2 dd . ddg XX vTNE tt (5.1) [4] 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. 205211. http://dx.doi.org/10.1016/0094114X(87)900036 Computing the scalar product of (5.1) with itself, we obtain the following equation for the geodesic curvature : [5] 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. 1522. http://dx.doi.org/10.1115/1.3171705 22 22 2 3 dd dd , dd dd . dd , dd g XX XX X tt tt XX tt [6] 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. 341347. http://dx.doi.org/10.1115/1.3267417 And a formula for geodesic torsion can be given as 23 22 dd d det,,,. ddd g g3 XX Ettt [7] M. P. Do Carmo, “Differential Geometry of Curves and Surfaces,” PrenticeHall, Englewood Cliffs, 1976, 503 p. This relation can be also seen in [2].
