A QUATERNIONIC APPROACH to GEOMETRY of CURVES on SPACES of CONSTANT CURVATURE

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1 INTERNATIONAL JOURNAL OF GEOMETRY Vol. 3 (2014), No. 1, A QUATERNIONIC APPROACH to GEOMETRY of CURVES on SPACES of CONSTANT CURVATURE TUNA BAYRAKDAR and A. A. ERG IN Abstract. We construct the Frenet-Serret frame for a curve on four dimensional Euclidean sace by means of quaternion algebra. To be able to generalize this aroach to curves on the other model saces of constant curvature we construct a quaternion algebra on each ber of the tangent bundle comatible to the Riemannian metric, i.e. algebra with the multilicative roerty with resect to the norm induced by the Riemannian metric. Via this construction we obtain Frenet-Serret frame and its derivative formulas for curves on three and four dimensional saces of constant curvature in the language of the quaternion algebra. 1. Introduction Any comlete, simly connected Riemannian manifold with constant sectional curvature C is locally isometric to one of the saces R n (C = 0); S n (C > 0) and H n (C < 0) for n 2, which are model saces for Euclidean and non- Euclidean geometries. The ullback metrics on these saces are given locally by (1) nx ds 2 E = (dx i ) 2 ds 2 S = i=1 ds 2 H = 1 (x n ) 2 4 (1 + (x 1 ) 2 + ::: + (x n ) 2 ) 2 nx (dx i ) 2 ; x n > 0; i=1 nx (dx i ) 2 where (x 1 ; :::; x n ) are coordinates on R n. The saces of constant curvature has an imortance in both theory and alications in di erential geometry and mathematical hysics. Keywords and hrases: Quaternions, Frenet-Serret Frame, saces of constant curvature. (2010)Mathematics Subject Classi cation: 53B20, 15A66 Received: In revised form: Acceted: i=1

2 54 Tuna Bayrakdar and A. Aziz Ergin In the context of di erential geometry, saces of constant(sectional) curvature has a central role to study geometric and toological roerties of a comlete Riemannian manifold by comaring its sectional curvature with that of a model sace [5]. Also in mathematical hysics, a sace of constant negative curvature aears (for examle) as a solution of nonlinear artial di erential equation of hyerbolic tye, namely sine-gordon equation, which naturally arises from the comatibility conditions on the Gauss equations for surfaces in 3-D with Gaussian curvature K = 1. Sine-Gordon equation describes a comletely integrable Hamiltonian system with in nitely many degrees of freedom, see for examle [3]. The theory of curves in classical di erential geometry is of secial interest esecially in three and four dimensions. In three dimension, a smooth curve is a vector-valued function of a single variable which is determined by its invariants, called curvature and torsion, u to a osition in the sace. The curvature and torsion describes deviation of a curve from its tangent line and oscillating lane resectively. The motion of moving triad along a curve described by the well known system of Frenet-Serret equations. Frenet- Serret equations for curves on n-dimensional Riemannian sace given rstly by Blaschke [2]. Quaternionic aroach to the curve theory in three and four dimensional Euclidean saces given in [1]. In that aer, authors achieved to obtain a frame for a curve in 4-D and its governing equations for a very secial choice. One of the objectives of this work is to construct Frenet frame in a general way and reresent its derivative formulas. Quaternion algebra is an elegant tool to study local roerties of an immersed geometric objects in a sense not only that makes the formulas simli ed but also serves a relevance that re ects a simle interlay between geometry and algebra. Invariants of a curve for examle, can be determined as a multilication of two quaternions due to the fact that interesting relation between scalar roduct and quaternion multilication. This serves that differential geometric roerties determined by the metric can be treated also in quaternion algebraic sense. Since the invariants(curvatures) are habitants of the center of the quaternion algebra, the class of non-congruent curves in four sace determined by taking quotient of quaternion algebra by its center, i.e curves obtained from non-congruent curves in three sace can not be congruent in four dimensions. This can be thought as a functional deendence criteria on curvatures. In this oint of view constructing a frame on a curve in four dimensions from the curve in three dimensions is geometrically meaningful. In this aer we deals essentially with the construction of a quaternion algebra on each ber of the tangent bundle of a sace form of dimensions 3 and 4 such that, the norm induced by the metric on a base sace has multilicative roerty, i.e. satis es jqj = jjjqj, for ; q 2 T x M: It is so the isomorhisms between tangent saces with algebraic structure on them determined by the metric structure on manifold. By means of this suitably arranged quaternion algebra we construct Frenet-Serret frame on curves in 3 and 4 dimensional saces of constant curvature and we calculate Frenet- Serret formulas for them. This occurs naturally as the generalization of the rocedure that we manage to frame a curve in three and four dimensional

3 Quaternionic Aroach to Geometry of Curves 55 Euclidean saces and we see that identities that we use during the construction of the Frenet-Serret frame become more simli ed thanks to the quaternion algebra. 2. Preliminaries 2.1. (Euclidean) Quaternions. Quaternions can be found in many texts [6],[9],[7]. The set H of quaternions q = a + bi + cj + dk is an associative, non-commutative division algebra over the reals with resect to the multilication (2) ij = k = ji; jk = i = kj; ki = j = ik; i 2 = j 2 = k 2 = 1; where 1 = (1; 0; 0; 0); i = (0; 1; 0; 0); j = (0; 0; 1; 0); k = (0; 0; 0; 1) are the basic vectors for R 4 and 1 stands for the unity of the algebra. The quaternion algebra is isomorhic to a subalgebra of two by two matrices with comlex entries M(2; C): (3) q = a + bi + cj + dk! a + bi c + di c + di a bi An oeration of conjugation of a quaternion q = a + bi + cj + dk is de ned by q = a1 bi cj dk: The corresonding ma q 7! q is an anti-isomorhism that is q 1 q 2 = q 2 q 1. As in the case of comlex numbers real(scalar) and imaginary(vectorial) art of a quaternion q = a + bi + cj + dk is de ned resectively by Re(q) = q + q 2 = a; Im(q) = bi + cj + dk: Thus any quaternion can be written in the form q = Re(q)+Im(q): The set of quaternions with vanishing real arts is given by ImH = fq 2 H : q = qg, which is identi ed with R 3. But ImH is not subalgebra of H. The Euclidean norm of a quaternion is given by jqj 2 = qq = a 2 + b 2 + c 2 + d 2 ; which is induced by the scalar roduct (4) < ; q >= Re(q) = 1 (q + q) : 2 With resect to this inner roduct any two quaternions are said to be orthogonal if q + q = 0: Any non-zero quaternion has an inverse q 1 = q jqj 2 : The set of all unit quaternions is just the 3-shere S 3 which isomorhic to SU(2), set of all unitary matrices with determinant 1.!

4 56 Tuna Bayrakdar and A. Aziz Ergin 3. Frenet-Serret Equations on R 3 To derive underlying equations for curves on R 4 recall the three dimensional case. Classical version of Frenet-Serret equations on R 3 can be found in [8]. Quaternionic aroach to this equations given in [1] is the following theorem: Theorem 3.1. Let q(s) = x(s)i+y(s)j +z(s)k be a unit seed regular curve on R 3 with > 0. Then the following formulas hold: (5) t 0 = n n 0 = t + b b 0 = n Here the functions and are curvature and torsion of given curve resectively. Using the identities b = tn, n = bt t = nb, the linear system of equations turns out to be the following system: (6) d ds t n b 1 A = b 0 0 0! t 1 0 t n b where! = b +t is Darboux quaternion. So the system of Frenet-Serret equations can be considered as a left quaternionic eigenvalue roblem [4],[10]. Integration of these rst order quaternion di erential equations is another story and it is not in the scoe of this aer. 1 A ; 4. Frenet-Serret Formulas on R 4 via Quaternions In this section we construct Frenet-Serret frame eld along a curve in four sace in virtue of a curve in three sace. We also give exlicit calculations of the rocedure and then we state the theorem describing motion of this frame along aforementioned curve. Let q(s) = (s) + (s)i + (s)j + (s)k be a unit seed regular curve in E 4, that is T = dq ; jt(s)j = 1 foralls: ds Let t; n; b denote resectively that the tangent, normal and binormal quaternions of a curve in three sace arametrized by an arc-length arameter l = l(s). The unit quaternion elds de ned by tt; nt; bt are erendicular to the quaternion T with resect to the Euclidean inner roduct (4) and hence the set ft; tt; nt; btg, forms an orthonormal basis for H w E 4. The rincial normal to the curve q = q(s) is de ned by (7) N := T0 jt 0 j ; jnj = 1; k 1 = jt 0 j > 0; where k 1 = jt 0 j is the rst curvature of q = q(s). Taking derivative of the both side of TT = 1, we see that T is orthogonal to N. Since N is a unit quaternion and erendicular to T, it can be written as 3X (8) N = c 1 tt + c 2 nt + c 3 bt; c 2 i = 1; c i = c i (s): i=1

5 Quaternionic Aroach to Geometry of Curves 57 Since P 3 i=1 c2 i = 1, _^c is orthogonal to the urely imaginary quaternion ^c = c 1 i + c 2 j + c 3 k, where dot denotes the derivative with resect to coordinate s. Let us denote the normalized quaternion along _^c by ^!: The quaternion de ned by _^c = ^!; 2 R; j!j = 1: (9) B 1 :=! 1 tt +! 2 nt +! 3 bt; 3X! 2 i = 1;! i =! i (s) is then orthogonal to N and T. Let us de ne the quaternion B 2 as i=1 (10) B 2 := B 1 NT: It is seen by a direct calculation that B 2 can be written exlicitly as (11) B 2 = (c 2! 3! 2 c 3 )tt + (c 3! 1 c 1! 3 )nt + (c 1! 2 c 2! 1 )bt The comonents of the quaternion B 2 in the basis tt; nt; bt are determined by the comonents of the urely imaginary quaternion ^a = ^c^! = (c 2! 3! 2 c 3 )i + (c 3! 1 c 1! 3 )j + (c 1! 2 c 2! 1 )k = a 1 i + a 2 j + a 3 k: Since ^c and ^! unit quaternions, B 2 is also unit quaternion: B 2 = a 1 tt + a 2 nt + a 3 bt; 3X a 2 i = 1; a i = a i (s): From de nition of T; N; B 1 ; B 2 it is easily seen that B 2? T, B 2? N, B 2? B 1, and thus the moving frame ft; N; B 1 ; B 2 g forms a orthonormal basis for R 4. Consequently, their derivatives with resect to arc-length arameter reresented by the skew-symmetric matrix due to the orthonormality relations: i=1 (12) T 0 = k 1 N N 0 = k 1 T + 1 B B 2 B 0 1 = 1 N + 2 B 2 B 0 2 = 2 N 2 B 1 : It is in fact more convenient to use the set moving quaternions ft; N; ^X 1 ; ^X 2 g, where ^X 1 = 1 k 2 ( 1 B B 2 ) ^X 2 = 1 k 2 ( 2 B 1 1 B 2 ) ; k 2 = q : One can easily see that ^X 1 is orthogonal to the subsace sanned by ft; N; ^X 2 g and ^X 2 is orthogonal to the subsace sanned by ft; N; ^X 2 g. Therefore, the set ft; N; ^X 1 ; ^X 2 g forms a basis for R 4, and consequently we obtain the

6 58 Tuna Bayrakdar and A. Aziz Ergin Frenet-Serret frame on a unit seed curve in R 4 via quaternion algebra: T = dq ds N = T0 k 1 ^X 1 = 1 k 2 1 B B 1 NT ^X 2 = 1 k 2 2 B 1 1 B 1 NT From (12) and the de nition of ^X 1 and ^X 2 we see that N 0 = k 1 T + k 2 ^X1 : Since ^X 1 is unit quaternion ^X 0 1? ^X 1 and from (12) ^X 0 1? T, so we have ^X 0 1 = 1 N + 2 ^X2 : Similarly, since ^X 2 is unit quaternion ^X 0 2? ^X 2 and from (12) ^X 0 2? T. It follows from here that ^X 0 2 = 1 N + 2 ^X1 : Since ^X 1? ^X 2, 2 = 2. Taking also derivative of the both sides of < N; ^X 2 >= 1 N^X ^X 2N = 0: gives 1 = 0. Taking again the derivative of the both sides of < N; ^X 1 >= 1 N^X ^X 1N = 0: results in 1 = k 2. Let us denote 2 by k 3. Thus we have roved the following theorem: Theorem 4.1. Let q(s) = (s) + (s)i + (s)j + (s)k be a unit seed curve in R 4 with curvature k 1 > 0, then the following formulas hold: (13) T 0 = k 1 N N 0 = k 1 T + k 2 ^X1 ^X 0 1 = k 2 N + k 3 ^X2 ^X 0 2 = k 3 ^X1 : Here k 1 ; k 2 and k 3 denote the rst, second and third curvatures resectively. In the roof of the theorem the functional deendence of curvatures with and have not aeared exlicitly. To see this one should reresent the derivatives of the frame ft; N; ^X 1 ; ^X 2 g in the basis T; tt; nt; bt and then comare the coe cients with k 2 and k 3. There is no condition on the tangent eld T so on k 1 by reason of that T is de ned indeendently from the frame t; n; b of the curve in R 3. To see that the curvature k 2, for examle, is a function of and calculate N 0 directly as (14) N 0 = dn ds = (c0 1 c 2 )tt+(c 0 2+c 1 c 3 )nt+(c 0 3+c 2 )bt k 1 T; where = dl ds. Since N0 is erendicular to N it can be written of the form N 0 = 1 B B T;

7 Quaternionic Aroach to Geometry of Curves 59 where 1 ; 2 ; 3 are de ned resectively by 1 = + a 3 + a 1 2 =! 3! 1 3 = k 1 ; q where = j _^cj. Since k 2 de ned as k 2 = , it is written as k 2 = f(; ). The condition on k 3 can be seen similarly. 5. Riemannian Quaternions: A Generalization of the Quaternion Algebra In this art of the work, we try to construct quaternion algebra in such a way that scalar roduct of given to quaternions with resect to Riemannian metric on base sace M arise naturally in the real art of the roduct of given two quaternions. More formally, we try to construct a quaternion algebra on each ber of the tangent bundle of M, such that the arc-length of a curve on M, with (0) = x, can be calculated as real art of the quaternion multilication of the tangent quaternion to the curve at oint x with its conjugate, i.e. ds 2 (15) = Re() = g ij i j ; 2 T x M; 0 (0) = : dt There are some obstructions to construct such an algebra that stems from the form of the metric on base sace. Let us start with the general form of the metric and then modify it to go further for our objectives. Let (M; g) be a four dimensional Riemannian manifold with Riemannian connexion r and let (x 1 ; x 2 ; x 3 ; x 4 ) be the local coordinates on M. In this coordinates distance element is written of the form (16) ds 2 = g ij dx i dx j : To construct of quaternion algebra on each ber of the tangent bundle with resect to quadratic form (16) we de ne the following multilication rule on the coordinate basis e i i g(e 2 ; e 2 ) e 2 e 2 = g(e1 ; e 1 ) e g(e2 ; e 2 )g(e 3 ; e 3 ) 1; e 2 e 3 = e 4 ; e 2 e 4 = g(e4 ; e 4 ) g(e2 ; e 2 )g(e 4 ; e 4 ) e 3 g(e3 ; e 3 ) e 3 e 2 = g(e3 ; e 3 )g(e 2 ; e 2 ) e 4 ; e 3 e 3 = g(e4 ; e 4 ) g(e 3 ; e 3 ) g(e1 ; e 1 ) e g(e3 ; e 3 )g(e 4 ; e 4 ) 1; e 3 e 4 = e 2 g(e2 ; e 2 ) e 4 e 2 = g(e4 ; e 4 )g(e 2 ; e 2 ) e 3 ; e 4 e 3 = g(e3 ; e 3 ) g(e4 ; e 4 )g(e 3 ; e 3 ) e 2 ; e 4 e 4 = g(e2 ; e 2 ) g(e 4 ; e 4 ) g(e1 ; e 1 ) e 1: Here e 1 is the unit element of the algebra and g ij = g(e i ; e j ). Accordingly, multilication of two quaternions = P i e i and q = P q i e i can be written in the form q = X i q j e i e j ; i;j where e i e j is de ned as above. With resect to this multilication rule conjugation oeration and taking inverse are de ned same as to classical

8 60 Tuna Bayrakdar and A. Aziz Ergin quaternion algebra. Analogous to Euclidean case we should exect the scalar roduct of two quaternions = P i e i and q = P q i e i with resect to the Riemannian metric on M is to be de ned by < ; q >= Re(q): If we calculate this with resect to the new quaternion algebra we nd (17) < ; q >= Re(q) = X i g ii i q i : g11 We see from (17) that to de ne the scalar roduct of two tangent quaternions as Re(q) it is needed only the diagonal elements of the metric tensor. Consequently, if one wants study the geometric objects via this quaternionic aroach, he should deal at least with the saces on which metric has a diagonal form i.e. ds 2 = g ii dx i dx i : For this tye of metrics it is always ossible to transform coordinates such that g 11 = 1 i.e. there exist a local coordinate system z i = z i (x 1 ; x 2 ; x 3 ; x 4 ) such that in this coordinates metric has the form (18) ds 2 = (dz 1 ) 2 + X g ii dz i dz i : i This coordinate system is referred coordinate system for us. In this sense, quaternion algebra which we want to construct is adatable with the metric of the form (18). In this coordinates, quadratic form (18) can be reresented by the quaternion multilication as ds 2 = Re(dqdq) = 1 (dqdq + dqdq) = dqdq; 2 and the scalar roduct of two quaternions can be interreted as (19) < ; q >= Re(q) = 1 2 (q + q) = X i i q i g ii : One can easily check that the norm induced by Riemannian metric (19) is comatible with quaternion algebra, i.e. it has multilicative roerty. In this oint of view, the saces of constant curvature are legitimate candidates to aly the quaternion aroach to them. In this sense, let us arrange the conformally at metrics on sace forms by a suitable change of coordinates to be able to investigate their submanifolds (only curves in this study) by the quaternion algebra which we constructed. Euclidean sace is automatically alicable with standard coordinate. Consider four dimensional hyerbolic sace with metric ds 2 H = 1 (x 4 ) 2 (dx1 ) 2 + ::: + (dx 4 ) 2 : Under the coordinate transformation z 1 = Z x 1 metric turns the following one: dx 4 (x 4 ) 2 ; z2 = x 2 ; z 3 = x 3 ; z 4 = x 4 ; (20) ds 2 H = (dz1 ) (z 4 ) 2 4X (dz i ) 2 : i=2

9 Quaternionic Aroach to Geometry of Curves 61 If we aly the similar coordinate transformation, metric on 4-shere becomes (21) ds 2 S = (dz1 ) (1 + (z 1 ) 2 + ::: + (z 4 ) 2 ) 2 4X (dz i ) 2 : Thus the quaternion algebras constructed from the metrics (20) and (21) are de ned resectively as i=2 e i e j = g(ei ; e i )g(e j ; e j ) " ijk e k ; i; j; k 6= 1 g(ek ; e k ) e i e 1 = g(e i ; e i )e 1 ; where " ijk is Levi-Civita tensor, g 11 = 1 and for the hyerbolic case and g ii = g ii = 1 (z 4 ; for i = 2; 3; 4; ) 2 4 (1 + (z 1 ) 2 + ::: + (z 4 ) 2 ; for i = 2; 3; 4; ) 2 for the sherical case. Corresonding quadratic forms are then evaluated by the simle quaternion multilication as (22) ds 2 = dqdq = (dz 1 ) 2 + g 22 (dz 2 ) 2 + g 33 (dz 3 ) 2 + g 44 (dz 2 ) 4 ; where g ii s are de ned as above. What the interesting art of this construction is that isomorhism between the algebras de ned on the tangent saces of corresonding manifolds determined by the metric structures on them Curves on Three Dimensional Saces. In this section we deal with Frenet-Serret equations for curves in three dimensional sace which is considered as the submanifold M of four dimensional sace of constant curvature ~M determined locally by the inclusion ma with vanishing rst comonent,say x 1 = 0, (we relaced z i by x i ). This submanifold is obviously isometrically embedded. In this case, the tangent sace of the submanifold is a subsace of urely imaginary quaternions constructed from the metric. In section 3, we touched on the Euclidean case, so we work only on non-euclidean cases. On submanifold M, Riemannian metric reduces to ds 2 = dqdq = g 22 (dx 2 ) 2 + g 33 (dx 3 ) 2 + g 44 (dx 2 ) 4 : Let x i ; i = 2; 3; 4 denote the (referred) local coordinates on M and let r be the induced connection on the submanifold M, determined by rojecting the Riemannian connection r ~ to the subsace and suose that the unit seed curve given in coordinates by x 2 (s); x 3 (s); x 4 (s). Since is unit seed curve its velocity vector t = t i e i satis es t i = dxi ds ; X t i t i g ii = 1:

10 62 Tuna Bayrakdar and A. Aziz Ergin Di erentiating P t i t i g ii = 1 gives 0 = d ds (ti t i g ii ) = dqk ds r k(t i t i g ii ) = dq k ds r kt i t i g ii = (r t t i )t i g ii Thus the covariant derivative of quaternion eld t is orthogonal to itself. (Here we use the notation d ds = dxk ds r k = r t for the derivative of a vector eld along a curve, where r k = r ek ). De ne the quaternion r t t jr t tj := n e 1, curvature can be deter- where jr t tj := is the curvature of the curve. Since n 2 = mined also by quaternion multilication: (23) (r t t)n := e 1 : Let us de ne the binormal quaternion as (24) b := tn; the quaternion multilication is done with resect to the table given in the beginning of this section. It is exlicitly found as b = (t 3 n 4 t 4 n 3 g33 g44 ) e 2 +(t 4 n 2 t 2 n 4 g22 g44 ) e 3 +(t 2 n 3 t 3 n 2 g22 g33 ) e 4 ; g22 g33 g44 where t k and n k s are the comonents of t and n. With a tedious calculations one can nd that jbj = 1 and also b is orthogonal to t and n. Since n and b are unit quaternions their covariant derivatives along the curve are orthogonal to theirselves: r t n = c 1 t + c 2 b; r t b = d 1 t + d 2 n; c i ; d i 2 R: It follows from here and (23) that r t (tn) = (r t t)n + t(r t n) = c 1 c 2 n: Since, r t (tn) = r t b = c 1 c 2 n has no t comonent, d 1 = 0. Besides that r t (tn) is urely imaginary quaternion, c 1 = holds. Let us de ne the torsion as (r t b)n := : It follows from here and from r t < n; b >= 1 2 r t(n b + bn) = 0 that c 2 =. Thus we have roved the following theorem: Theorem 5.1. Let (s) be a unit seed curve on one of three dimensional Riemannian sace of constant curvature with 6= 0, then the following formulas hold: (25) r t t = n r t n = t + b r t b = n

11 Quaternionic Aroach to Geometry of Curves Curves on Four Dimensional Sace In this last section of the work we construct Frenet-Serret frame on a curve on four dimensional Riemannian saces of constant curvature H 4 and S 4 analogous to technique that we have followed for a curve on R 4. We construct Frenet-Serret frame on a unit seed curve in four dimensional sace ~ M with the hel of the Frenet frame of a curve in three-dimensional submanifold M, determined by x 1 = 0, with resect to the metric from which we construct a generalized quaternion algebra. Let t; n; b denote resectively that the tangent, normal and binormal quaternions of a curve in M arametrized by its arc-length l = l(s) and let T denotes the tangent of the unit seed curve = (s), given locally by the (x 1 (s); x 2 (s); x 3 (s); x 4 (s)): T i = dxi ds ; Ti T i g ii = 1: The unit quaternions tt; nt; bt in T ~ M are erendicular to the quaternion T with resect to the metric (22). Indeed, let T = T i e i for i = 1; 2; 3; 4 and t = t i e i for i = 2; 3; 4 be given in coordinate basis. We will show that g ii T i (tt) i = 0: One can nd by a direct calculation that tt = + + 4X i=2! t i T i g ii e 1 + t 2 T 1 + (t 3 T 4 t 4 T 3 g33 g44 ) e 2 g22 t 3 T 1 + (t 4 T 2 t 2 T 4 g22 g44 ) e 3 g33 t 4 T 1 + (t 2 T 3 t 3 T 2 g22 g33 ) e 4 : g44 Therefore, it is easily nd that g ii T i (tt) i = Re T(tT) = 1 2 T(tT) + (tt)t = 0; That is, tt? T. We see here that the comutations in coordinates become very simli ed as a single quaternion equation. Reeating similar calculations for nt; bt results in nt? T and bt? T, and hence the set ft; tt; nt; btg of quaternions, forms an orthonormal basis for tangent sace of the Riemannian manifold M ~ at a given oint. Analogous to Euclidean case we de ne the quaternions T = d ds N = r ~ T j r ~ T j = c 1tT + c 2 nt + c 3 bt B 1 =! 1 tt +! 2 nt +! 3 bt B 2 = B 1 NT = a 1 tt + a 2 nt + a 3 bt;

12 64 Tuna Bayrakdar and A. Aziz Ergin where the coe cients are reresented by the urely imaginary quaternions ^c = c 2 e 2 + c 3 e 3 + c 4 e 4 ^! =! 2 e 2 +! 3 e 3 +! 4 e 4 ;! i = _^c i j _^cj ^a = ^c^! = X gii gjj (c i! j! i c j ) e k ; gkk i<j;k6=i;j quaternion multilications are imlemented by the metric. It is easily seen that the set ft; N; B 1 ; B 2 g of quaternions forms an orthonormal basis. So their derivatives with resect to the arc-length arameter s reresented by skew-symmetric matrix due to the orthonormality, that is, (26) ~r T T = k 1 N De ne quaternions ^X 1 and ^X 2 as ~r T N = k 1 T + 1 B B 2 ~r T B 1 = 1 N + 2 B 2 ~r T B 2 = 2 N 2 B 1 : ^X 1 = 1 k 2 ( 1 B B 2 ) ^X 2 = 1 k 2 ( 2 B 1 1 B 2 ) ; k 2 = g(^x 1 ; ^X 1 ) = g(^x 2 ; ^X 2 ); here k 2 = g(^x i ; ^X i ) = q It will again be more convenient to use the quaternions T; N; ^X 1 ; ^X 2 as an orthonormal frame along curve. From the orthogonality relations one can easily see that ~ r T ^Xi, for i = 1; 2 has no tangential comonent and ~ r T ^X2 has no normal comonent. If we denote the comonent of ~ r T ^X2 in the direction of ^X 1 by k 3, then the system of equations (26) turns out to be the following system: ~r T T = k 1 N ~r T N = k 1 T + k 2 ^X1 ~r T ^X1 = k 2 N + k 3 ^X2 ~r T ^X2 = k 3 ^X1 ; thus we have roved the following theorem. Theorem 6.1. Let = (s) be a unit seed curve in four dimensional sace of constant curvature with non-vanishing rst curvature, then the following formulas hold: 0 (27) rt ~ T N ^X 1 ^X 2 1 C A = 0 where k 1 ; k 2 and k 3 de ned as above 0 k k 1 0 k k 2 0 k k B A T N ^X 1 ^X 2 1 C A :

13 Quaternionic Aroach to Geometry of Curves Concluding Remarks and the Following Objectives By means of the quaternion algebra, we able to construct Frenet-Serret frame on a curve in R 4. This construction is achieved by a curve in three sace in a sense that second and third curvature of the curve in R 4 determined by and. To be able to aly this aroach to curves on a three and four dimensional Riemannian saces of constant curvature, we construct a generalized quaternion algebra on each ber of tangent bundle in such a way that multilication rule determined by the comonents of the metric tensor. Identical to the standard quaternion algebra, inner roduct of two quaternions ; q in the tangent bundle aears as the real art of multilication q. Eventually, we construct an orthonormal frame on a curve on the model saces algebraically via this generalized quaternion algebra. We see that with this construction, calculations deending on the metric tensor are rather simli ed. 8. Acknowledgements This aer was artially suorted by the Research Project Administration of Akdeniz University with roject number References [1] Bharathi, K. and Nakaraj M., Quaternion Valued Function of a Real Serret-Serret- Frenet Formula, Indian Journal of Pure and Alied Mathematics, 18(6)(1987), [2] Blaschke, W., Frenets Formeln für den Raum von Riemann, Math. Z., 6(1920), [3] Bobenko, A.I. and Seiler, R., Discrete Integrable Geometry and Physics, Oxford University Press, [4] Brenner, J.L., Matrices of Quaternions, Paci c J. Math., 1(1951), [5] Cheeger, J. and Ebin, D.G., Comarison Theorems in Riemannian Geometry, North- Holland, [6] Dubrovin, B., Novikov, S.P. and Fomenko, A.T., Modern Geometry Methods and Alications (Second Addition), Sringer-Verlag, [7] Kantor, I. L. and Solodovnikov, A.S., Hyercomlex Numbers An Elementary Introduction to Algebras, Sringer-Verlag, [8] O neill, B., Elementary Di erential Geometry, Academic Press, [9] Ward, J. P., Quaternions and Cayley Numbers: Algebra and Alications, Kluwer Academic Publishers, 403(1997). [10] Zhang, F., Quaternions and matrices of quaternions, Linear Algebra Al., 251(1997), AKDEN IZ UNIVERSITY DUMLUPINAR BOULEVARD ANTALYA, TURKEY. address: tunabayrakdar@akdeniz.edu.tr aaergin@akdeniz.edu.tr