RULED SURFACES WITH CONSTANT SLOPE RULING ACCORDING TO DARBOUX FRAME IN MINKOWSKI SPACE

In this study, three different types of ruled surfaces are defined. The generating lines of these ruled surfaces are given by points on a curve X in Minkowski Space, while the position vector of X have constant slope with respect to the planes (t, y) , (t, n) , (n, y). It is observed that the Lorentzian casual characters of the ruled surfaces with constant slope can be timelike or spacelike. Furthermore, striction lines of these surfaces are obtained and investigated under various special cases. Finally, new investigations are obtained on the base curve of these types of ruled surfaces.


Introduction
A ruled surface is a special surface which is formed by moving a line along a given curve in 3-dimensional Minkowski space. The line is called the generating line and the curve is called the direction curve of the surface. Thus, a ruled surface has a parametrization M (u, v) = α(u) + vX (u) where α and X are curves. The curve α is called the directrix or base curve and X is called the director curve of the ruled surface. Thus, the ruled surfaces in Minkowski space can be classified according to the Lorentzian character of their ruling and surface normal. Developable ruled surfaces are surfaces which can be made isometric to part of the plane. The necessary and sufficient conditions for these surfaces to become developable are characterized by vanishing Gaussian curvature. In this study "developable" and "torsal" are used as synonyms, since a surface is developable if and only if it is a torsal ruled surface. Cylindrical, conical, torse surfaces, a plane and surfaces of polyhedrons are examples of torsal surfaces. These surfaces can be developed on a plane without any lap break. On the ground that their isometrics with planes are becoming interesting to discover more ways to use these surfaces in different applications in [5] .
A regular curve in E 3 1 , whose position vector is obtained by Frenet frame vectors on another regular curve, is called Smarandache curve [1]. In this study, it is shown that if developable surface's generating line is a Smarandache curve and asymptotic or geodesic curve, then the basic curve is a general helix.
K. Malecek and others defined the surfaces with a constant slope with respect to the given surface in Euclidean Space in [7]. At the same time in [10], Yavuz, Ateş and Yaylı investigated surface with a constant slope ruling with respect to osculating plane by using Frenet Frame according to casual characters in Minkowski space. By the definition of surfaces with a constant slope ruling with respect to the given surface in study [7] , in this study new surface definitions are obtained. These surfaces in three types were studied according to the Darboux frame in Minkowski Space. Furthermore, necessary and sufficient conditions are given for these surfaces to become developable in Minkowski 3-space. Striction lines of the surfaces are obtained and investigated under various special cases. Finally, the ruled surfaces with constant slope ruling visualized of given curves as examples, separately.

Preliminaries
A curve in a manifold M is a smooth mapping α : I → M, where I is an open interval in the real line R.
A curve α in a semi-Riemannian manifold M is spacelike if all of its velocity vectors α (s) are spacelike, is null if all of its velocity vector α (s) are null, timelike if all of its velocity vectors α (s) are timelike [8].
In this study, the Darboux frames and formulas in the Minkowski space E 3 1 are given with metric Let S be an oriented surface in E 3 1 and let consider a non-null curve α (s) lying fully on S . Since the curve α (s) lies on the surface S there exists a frame along the curve α (s). This frame is called Darboux frame and denoted by {t, y, n} which gives us an opportunity to investigate the properties of the curve according to the surface. In this frame t is the unit tangent of the curve, n is the unit normal of the surface S along curve α (s) and y is a unit vector given by y = ∓n × t. According to the Lorentzian casual characters of the surface and the curve lying on surface, the derivative formulae of the Darboux frame can be changed as follows: i) If the surface is timelike, then the curve α (s) lying on surface can be spacelike or timelike. Thus, the derivative formulae of the Darboux frame is given by ii) If the surface is spacelike, then the curve α (s) lying on surface is spacelike. Thus, the derivative formulae of the Darboux frame is given by where t, t = 1, y, y = 1, n, n = −1 Here, is the geodesic curve, k n is the normal curvature defined by equality k n (s) = α (s) , n (s) and t r is the geodesic torsion of α(s) defined by t r (s) = − n (s) , y (s) [2], [3], [4].
If u, v ∈ E 3 1 , Lorentzian vector product of u and v is to the unique vector by u × v that satisfies where u × v vector product is defined as follows The relations between geodesic curve, normal curvature, geodesic torsion and κ and τ are given, if both surface and curve are timelike or spacelike, then if surface is timelike and curve is spacelike, then , [3], [4].
Let − → x and − → y be future pointing (or past pointing) timelike vectors in R 3 1 . Then there is a unique real number θ > 0 such that Let − → x and − → y be spacelike vectors in R 3 1 that span a timelike vector subspace. Then there is a unique real number θ > 0 such that Let − → x and − → y be spacelike vectors in R 3 1 that span a spacelike vector subspace. Then there is a unique real number θ > 0 such that Let − → x be a spacelike vector and − → y be a timelike vector in R 3 1 . Then there is a unique real number θ > 0, such that − → x , − → y = − → x − → y sinh θ. [9].

3-Space
Let M be a ruled surface whose generating lines are given by points on the curve X in Minkowski Space , while in all points they have the constant slope with respect to the tangent planes to the given surface Case 3.3. If α(s) is a timelike curve with the principal spacelike normal vector field n(s), then surface is timelike where σ 2 + 1 > 0 and direction vector is given as follows so the timelike surface is parametrized by where the vector t is the direction vector of a tangent to the curve X, n is the direction vector of a normal to the surface and y = n × t is the direction vector of intersection line of a tangent plane to the surface and the normal plane curve X at the point.
Proof. The surface M 1 is developable if and only if det(t, X, X ) = 0. Thus derivative of the direction vector of generating lines of the surface is obtained as follows And Proof. If X(s) is asymptotic curve, then k n = 0 and if surface and curve are the same character, then κ 2 = k 2 n + k 2 g , so k g = κ, t r = τ . If we replace the values in the last equation, we get the following equality

s generating line X(s) is a geodesic and Smarandache curve if and only if α(s)
is a general helix, so that where sinw(s) = x 1 = const.and cosw(s) = x 2 = const.
if also u(s) is a geodesic, then base curve is a general helix, so that Corollary 3.4. Let M 2 (s, v) be a spacelike developable surface and if u(s) is a line and Smarandache curve, where sinhw(s) = x 3 = const.and coshw(s) = x 4 = const. if at the same time u(s) be a asymptotic, then base curve is a general helix, so that if also u(s) is a geodesic, then base curve is a general helix, so that Proof. If M 2 (s, v) is a timelike developable surface and the base curve is spacelike, then κ 2 = k 2 g − k 2 n . So we replace values k n = 0, k g = κ, t r = τ in condition of developable equation ε sinh w(s) (sinh w(s).t r ) = σ (cosh w(s) (k g ) + σt r ) If u(s) is a geodesic, then we write values k g = 0, k n = −κ, t r = τ in condition of developable equation for the timelike surface, we obtained as follows where sinhw(s) = x 3 = const.and coshw(s) = x 4 = const.
Theorem 3.4. The striction line on spacelike surface M 2 (s, v) is given by where σ 2 − 1 > 0, and striction line on timelike surface is given by Remark 3.4. If u(s) is a geodesic curve and where σ 2 − 1 > 0, then striction line of spacelike surface is equal to base curve.  at the same time δ(s) be a asymptotic, then the base curve is a general helix, so that τ κ = σx 3 εx 2 4 − σ 2 , if also δ(s) be a geodesic, then the base curve is a general helix, so that where sinhw(s) = x 3 = const.and coshw(s) = x 4 = const.
Theorem 3.6. The striction line on timelike surface M 3 (s, v) is obtained as follows Case 3.6. If α(s) is a timelike curve with the principal spacelike normal vector field n(s), then the surface is timelike where σ 2 + 1 > 0 and direction vector is given as follows δ(s) = sinh w(s).t(s) + cosh w(s).n(s) + σ.y(s).
if also u(s) be a geodesic, then the base curve is a general helix, so that Remark 3.10. If sinhw(s) = x 3 = const.and coshw(s) = x 4 = const,generating lines of the surface u(s) be a Smarandache curve. Let M 2 (s, v) be a developable timelike surface and u(s) be a Smarandache curve , if at the same time u(s) be an asymptotic, then the base curve is a general helix with if also u(s) be a geodesic, then the base curve is a general helix with Theorem 3.10. The striction line on spacelike surface M 3 (s, v) is given by where σ 2 − 1 > 0, and striction line on timelike surface is given by w (s) 2 +k 2 n −2.k n .w (s) sinh 2 w(s) − ε cosh 2 w(s) +σ 2 k 2 g +t 2 r +2σ (k g sinh w (s) . (w (s) + k n ) +ε.t r cosh w (s) . (w (s) − εk n )) Remark 3.11. If u(s) is an asymptotic curve and where σ 2 − 1 > 0, then striction line of spacelike surface is equal to the base curve and where σ 2 − 1 < 0, then striction line of timelike surface is equal to the base curve.  if at the same time δ(s) be a asymptotic, then basic curve is a general helix, so that if also δ(s) be a geodesic, then base curve is a general helix, so that if δ(s) be a line curvature, then such that ε = t, t = ∓1.
Theorem 3.11. The striction line on timelike surface M 3 (s, v) is given by  The surface with constant slope M 1 is parametrized by  The surface parametrization is given as follows where sinw(s) = x 1 = const.and cosw(s) = x 2 = const.

Some Numerical Examples
In this section, we give examples of the surfaces with a constant slope ruling according to Darboux frame in Minkowski Space with respect to the given planes. Surface is visualized in following figure for w(s) = π 2 , r = 10, σ = 2.

Conflicts of Interest:
The author(s) declare that there are no conflicts of interest regarding the publication of this paper.