THE FRENET VECTOR FIELDS AND THE CURVATURES OF THE NATURAL LIFT CURVE

NATURAL LIFT CURVE Evren ERGÜN, Mustafa BİLİCİ and Mustafa ÇALIŞKAN 1 Ondokuz Mayıs University Faculty of Arts and Sciences Department of Mathematics,Samsun, Turkey. 2 Ondokuz Mayıs University Educational Faculty Department of Mathematics,Samsun, Turkey 3 Gazi University Faculty of Sciences Department of Mathematics, Ankara, Turkey. 1 eergun@omu.edu.tr, 2 mbilici@omu.edu.tr, 3 mustafacaliskan@gazi.edu.tr

Abstract: In this paper, Frenet vector fields, curvature and torsion of the natural lift curve of a given curve is calculated by using the angle between Darboux vector field and the binormal vector field of the given curve in . Also, a similar calculation is made in considering timelike or spacelike Darboux vector field.

1.Introduction
Thorpe gave the concepts of the natural lift curve and geodesic spray in [7]. Thorpe provied the natural lift  of the curve  is an integral curve of the geodesic spray iff  is an geodesic on M in [7].Çalışkan, Sivridağ and Hacısalihoğlu studied the natural lift curves of the spherical indicatries of tangent, principal normal, binormal vectors and fixed centrode of a curve in [2].They gave some interesting results about the original curve were obtaied, depending on the assumption that the natural lift curve should be the integral curve of the geodesic spray on the tangent bundle T(S 2 ) in [2]. Ergün and Çalışkan defined the concepts of the natural lift curve and geodesic spray in Minkowski 3-space in [6].The anologue of the theorem of Thorpe was given in Minkowski 3-space by Ergün and Çalışkan in [3].Walrave characterized the curve with constant curvature in Minkowski 3-space in [8].
Let be regular curve with arclength parameter s. , is the standard scalar product of given by for each vectors The norm of a vector is defined by Let be a parametrized curve. We denote by the moving Frenet frame along the curve , where T, N and B are the tangent, the principal normal and the binormal vector of the curve , respectively.
Let  be a reguler curve in . Then Let  be a unit speed space curve with curvature K and torsion , then where D is the Levi-Civita connection on , [7].
Let Minkowski 3-space be the vector space equipped with the Lorentzian inner product g given by where A vector is said to be timelike if g(X,X)<0, spacelike if g(X,X)>0 and lightlike (or null) if (X,X)=0. Similarly, an arbitrary curve where t is a pseudo-arclength parameter, can locally be timelike, spacelike or null (lightlike), if all of its velocity vectors (t) are respectively timelike, spacelike or null (lightlike), for every A lightlike vector X is said to be positive (resp. negative) if and only if x 1 >0 (resp. x 1 <0) and a timelike vector X is said to be positive (resp. negative) if and only if x 1 >0 (resp. x 1 <0). The norm of a vector X is defined by Let  be a unit speed spacelike space curve with a spacelike binormal. In this trihedron, we assume that T and B are spacelike vector fields and N is a timelike vector field In this situation,.
Then, Frenet formulas are given by Let  be a unit speed spacelike space curve with a spacelike binormal. In this trihedron, we assume that T and N are spacelike vector fields and B is a timelike vector field.In this situation,.
Then, Frenet formulas are given by, Lemma 1: Let X and Y be nonzero Lorentz orthogonal vectors in . If X is timelike, then Y is spacelike, [6]. Lemma 2: Let X and Y be pozitive (negative ) timelike vectors in . Then whit equality if and only if X and Y are linearly dependent. Lemma 3: (i) Let X and Y be pozitive (negative ) timelike vectors in . By the Lemma 2, there is unique nonnegative real number (X,Y) such that the Lorentzian timelike angle between X and Y is defined to be X,Y. ii) Let X and Y be spacelike vektors in that span a spacelike vector subspace. Then we have Hence, there is a unique real number X,Ybetween 0 and π such that the Lorentzian spacelike angle between X and Y is defined to be X,Y.
(iii) Let X and Y be spacelike vectors in that span a timelike vector subspace. Then we have Hence, there is a unique pozitive real number X,Ybetween 0 and π such that the Lorentzian timelike angle between X and Y is defined to be X,Y.
(iv) Let X be a spacelike vector and Y be a pozitive timelike vector in Then there is a unique nonnegative reel number X,Ysuch that the Lorentzian timelike angle between X and Y is defined to be X,Y, [6]. , then W is a timelike vector. In this situation, from Lemma 3 iv) we can write ii) For the curve  with a timelike principal normal, being an angle between the B and the W, if B and W spacelike vectors that span a spacelike vector subspace then by the Lemma 3 ii) we can write iii) For the curve with a timelike binormal, being a Lorentzian timelike angle between the -B and the W, a) If , then W is a spacelike vector. In this situation, from Lemma 3 iv) we can write Let M be a hypersurface in and let be a parametrized curve. is called an integral curve of X if where X is a smooth tangent vector field on M, [5]. We have where T P M is the tangent space of M at P and Mis the space of vector fields on M.
For any parametrized curve , given by is called the natural lift of on TM. Thus, we can write where D is the Levi-Civita connection on , [3]. For any parametrized curve in , given by is called the natural lift of on TM, [7].
We denote by the moving Frenet frame along the curve  , where T , is called the natural lift of on TM, [3].
We denote by the moving Frenet frame along the curve  , where T , N and B are the tangent, the principal normal and the binormal vector of the curve  , respectively. Corollary3: Let be a unit speed timelike space curve and  be the natural lift of .If W is a spacelike vector field, then Corollary4: Let be a unit speed timelike space curve and the natural lift  of the curve be a space curve with with curvature and torsion . If W is a spacelike vector field, then Corollary5: Let be a unit speed timelike space curve and  be the natural lift of .If W is a timelike vector field, then Corollary6: Let be a unit speed timelike space curve and the natural lift  of the curve be a space curve with curvature and torsion . If W is a timelike vector field, then Corollary7: Let be a unit speed spacelike space curve with a spacelike binormal and  be the natural lift of .Then Corollary8: Let be a unit speed spacelike space curve with a spacelike binormal and the natural lift  of the curve be a space curve with curvature and torsion . Then Corollary9: Let be a unit speed spacelike space curve with a timelike binormal and  be the natural lift of . If W is a spacelike vector field, then Corollary10: Let be a unit speed spacelike space curve with a timelike binormal and the natural lift  of the curve be a space curve with curvature and torsion . If W is a spacelike vector field, then

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BSMaSS Volume 2 Corollary11: Let be a unit speed spacelike space curve with a timelike binormal and  be the natural lift of . If W is a timelike vector field, then Corollary12: Let be a unit speed spacelike space curve with a timelike binormal and the natural lift  of the curve be a space curve with curvature and torsion . If W is a timelike vector field, then