The Frenet and Darboux Instanfaneous Rotation Vectors of Curves on Time-like Surface

In this paper, depending on the Darboux instantaneous rotation vector of a solid perpendicular trihedron in the Minkowski 3-space R j 3 =[ R 3 , (+,+,-)] the Frenet instantaneous rotation vector was stated for a space-like curve (c) with the principal normal n being a time-like vector. The Darboux instantaneous rotation vector for the Darboux trihedron was found when the curve (c) is on a time-like surface. Some theorems and results giving the relations between two frames were stated and proved.


INTRODUCTION
In Euclidean Space R 3 , a solid perpendicular trihedrons' Darboux instantaneous rotation vector and Darboux and Frenet instantaneous rotation vectors of a curve on the surface are known.Let <p be the angle between principal normal n and surface normal N on a point P of the curve.For the radii of geodesic torsion T g , normal curvature R n and geodesic curvature R g , some relations are given in [1].
i) (a,a»O then a is space-like vector, ii) < a , a) < 0 then a is time-like vector, iii)( a , a ) = 0 then a is light-like(null) vector.
The vectoral product of a and b is given by Defmition!.4 Let y= y(u,v) be a surface in space Ri.If 'if pe y(u,v) and ( , > I, is a Lorentzian metric then y(u, v) is time-like surface [5]. Def"mition1.
It can be written

DARBOUX TRIHEDRON FOR A SPACE-LIKE CURVE WITH TIME-LIKE GEODESIC NORMAL
Let us consider time-like surface y = y(u,v).For a space-like curve (c) on y = y(u,v) there exists the Frenet trihedron [t, n , b] at all points of (c) .There is also a second trihedron because the curve (c) is on surface y = y(u,v).Let us denote the tangent space-like unit vector with t , the normal space-like unit vector with N at the point P .In this case, tfwe take the time-like vector g defined by then we obtain a new trihedron [ t , g , N] .
Let e is the hyperbolic angle between the time-like vectors Nand g (Figurel).
If the both sides of this equation are scalarly multiplied with the vectors N and g and considered the equalities ( N , g ) = 0 , (0, g )= -cosh8 , (g, g )= -1 , the proof is completed.

5
Let a = ( al , a2 ) and b = ( bl , b2 ) ER~be future-pointing ( past-pointing) time-like vectors.The number e ERin equality r coshB sinhB l[ al 1=r bl ] L-sinh 0 coshB J b l J lb 2 in this equality eeR number is an hyperbolic angle from a to b and e is shown by e = (a, b )[2].Lemma 1.6 Let a and b be future pointing time-like unit vectors.If e is an hyperbolic angle from.a and b, then Let [el,e2,e3] be a solid perpendicular trihedron in space R:.In this situation the following theorem can be given: Theorem.!.7 If a solid perpendicular trihedrons' unit vectors el, e2 e3 are changing relative to t parameter, and e2 are space-like vectors, e3 IS a time-like vector and Darboux instantaneous rotation vector is 2. FRENET TRIBEDRON FOR A SPACE-LIKE SPACE CURVE WITH TIME-LIKE PRINCIPAL NORMAL Let us consider c = c(s) space-like curve.For any parameter s on all points on this curve, we can construct Frenet trihedron (t , n ,b), here,t ,n and b tangent ,principal normal and binormal unit vectors, respectively.In this trihedron, 0 is timelike unit ve(..10r;t and b are space-like unit vectors.For that, (t,1)= (b,b)=I, (0,0)=-1 tit ds = au t + a12 0 + alJ b do -= a 21 t + a 22 0 + a 23 b ds db ds = a31 t + ano + a33 b for a parameter's specific value s = So with t = t (s) , 0 = o(s) ,and b = b(s).Ifwe take derivatives of eqs.(2.1) and (2.2) with respect to arc s ,then we have dt db do (t, ds) = (b,ds ) = (o,ds ) = 0 ili ~ili ~~( ds,o)+(t, ds)=(ds' b)+(t'ds)=(ds' b)+(o'ds)= 0 we obtain the following formulas for derivatives : tit ds do ds db dsOn a space-like curve c = c(s) given a point P, if the radius of curvature is R and radius of torsion is T then Frenet formulae are obtained as following[7].and (2.7) are compared, then we obtain c = 1 / R , b = 0 , a = 1 / 1'.For that, for any perpendicular trihedron ,In this situation, if we apply the formula (1.5) obtained for a general perpendicular trihedron to Frenet trihedron, we can write

iP 2
ITwe take the derivatives of t , g and N according to the arc s of CUlVe(c) , we can find dt t sinh e + (T + ds ) derivative formulae are given by dt tis = Pg g -Pn N dg ds = Pg t + 'r g N dN ds = Pn t + r g g The matrix form of (3.3) is r t 1 [0 P g ~l~r P, 0 derivative formulae (3.3) can be given with Darboux vector as below dt =Wl\t ds dg = W 1\ g .ds tiN -= W 1\ N ds Theorem 3.1 If the radius of torsion of the space-like (c) drawn on time-like surface y = y(u, v) is T and the hyperbolic angle between time-like unit vectors D andg is e, then we we take derivative of both sides of equation < D , g) = -coshS and use the (2.7) and (3.3) we obtain tin g + n dg = _sinhB dB , g ) + T ( b, g ) + R (0, t ) If the radius of curvature of the space-like curve (c) on timelike surface y = y(u,v) is R and the hyperbolic angle between time-like unit vectors 0 and g is e.

Corollary 3 . 3 . 4 .
There is a relation between Frenet and Darboux vectors as follows: Instantaneous rotation velocity vector w of the Darboux trihedron is consists of two components.One of them coincides with Frenet trihedrons' instantaneous rotation velocity vector.The other is a component in the opponent direction of tangent and equals to de/ds.For this reason, when a point P of space-like curve (c) on the time-like surface moves on this curve Darboux trihedron moves with radial velocity de / ds according to Frenet trihedron in the opposite direction to the tangent at any time.Corollary 3.5  If the hyperbolic angle between the principal normal of a spacelike curve on a time-like surface and the vector g at the same of point of the surface of g vector becomes always constant, torsion of the curve at each point equals to the geodesic torsion of the surface at that point.