Non-null Slant Ruled Surfaces

In this study, we define some new types of non-null ruled surfaces called slant ruled surfaces in the Minkowski 3-space E_1^3. We introduce some characterizations for a non-null ruled surface to be a slant ruled surface in E_1^3. Moreover, we obtain some corollaries which give the relationships between a non-null slant ruled surface and its striction line in E_1^3.


Introduction
In the study of curve theory, the curves whose curvatures satisfy some special conditions have an important role. The well-known of such curves is general helix defined by the classical definition that the tangent lines of the curve makes a constant angle with a fixed straight line [4]. In 1802, M.A. Lancret stated a result on the helices which was first proved by B. de Saint Venant in 1845 [20]. Venant showed that a curve is a general helix if and only if the ratio of the curvatures κ and τ of the curve is constant, i.e., / κ τ is constant at all points of the curve. Helices have been studied not only in Euclidean spaces but also in Lorentzian spaces by some mathematicians and different characterizations of these curves have been obtained according to the properties of the spaces [6,7,10,15]. Recently, Izumiya and Takeuchi have introduced a new curve called slant helix which is defined by the property the normal lines of the curve make a constant angle with a fixed direction in Euclidean 3-space 3 E [8]. Later, the spherical images, the tangent indicatrix and the binormal indicatrix of a slant helix have been studied by Kula and Yaylı and they have obtained that the spherical images of a slant helix are spherical helices [11]. The position vector of a slant helix in 3 E has been studied by Ali [3]. Then the corresponding characterizations for the position vector of a timelike slant helix in Minkowski 3-space 3 1 E have been given by Ali and Turgut [2]. Recently, Ali and Lopez have also given some new characterizations of slant helices in Minkowski 3-space 3 1 E [1]. Analogue to the curves, ruled surfaces have orhonormal frames along their striction curves. So, the notion "helix" or "slant helix" can be considered for ruled surfaces. Before, Önder and Kaya have studied this subject for null scrolls and defined slant null scrolls in 3 1 E [19]. In this paper, we define non-null slant ruled surfaces by considering the Frenet vectors of timelike and spacelike ruled surfaces in 3 1 E . We give the conditions for a non-null ruled surface to be a slant ruled surface.

Preliminaries
Let 3 1 E be a Minkowski 3-space with natural Lorentz Metric  E . Then we have the following parametrization for a ruled surface N , The straight lines of the surface are called rulings and the curve ( ) k k u = is called base curve or generating curve. In particular, if the direction of q is constant, the ruled surface is said to be cylindrical, and non-cylindrical otherwise.
The function defined by , , , dk q dq dq dq δ = is called the distribution parameter (or drall) of the ruled surface. Then, N is called developable surface if and only if 0 δ = [12,16,18]. Then at all points of same ruling, the tangent planes are identical, i.e., tangent plane contacts the surface along a ruling. If , , 0 dk q dq ≠ , then the tangent planes of the surface N are distinct at all points of same ruling which is called nontorsal [16,18].
It is clear that the base curve is the same with striction curve if and only if , 0 dq dk = .
Since the vectors a and q are orthogonal, we can define an orthonormal frame on the surface. For this purpose, let write h a q = × . The unit vector h is called central normal and the orthonormal frame { } ; , , C q h a at central point C is called Frenet frame of N .
According to the Lorentzian casual characters of ruling and central normal, the Lorentzian character of the surface N is classified as follows; i) If the central normal vector h is spacelike and q is timelike, then the ruled surface N is said to be of type N − .
ii) If the central normal vector h and the ruling q are both spacelike, then the ruled surface N is said to be of type N + .
iii) If the central normal vector h is timelike, then the ruled surface N is said to be of type N × [16,18].
The ruled surfaces of type N − and N + are clearly timelike and the ruled surface of type N × is spacelike. By using these classifications and taking the striction curve as the base curve the parametrization of the ruled surface N can be given as follows, where , and ii) If the ruled surface N is spacelike ruled surface then we have In the equations (4) and (5) 3 1 E In this section, we introduce the definition and characterizations of q -slant ruled surfaces in 3 1 E . First, we give the following definition.

q -Slant Ruled Surfaces in
where ( ) c s is striction curve of N and s is arc length parameter of ( ) c s . Let the Frenet frame and non-zero invariants of N be { } , , q h a and 1 2 , k k , respectively. Then, N is called a q -slant ruled surface if the ruling ( ) q s makes a constant angle with a fixed non-null unit direction u in the space, i.e., , Then we give the following characterizations for q -slant ruled surfaces in 3 1 E . Whenever we talk about N we will mean that the surface has the properties as assumed in Definition 3.1. 1 2 / k k is constant and given by

Theorem 3.1. The ruled surface N is a q -slant ruled surface if and only if the function
where q c and a c are real constants. By differentiating (10) with respect to s it follows and then we have that the function Conversely, given a non-null ruled surface N , the equation (8) is satisfied. We define q a u c q c a = + , where , , , q a q u c a u c = = are non-zero constants. Differentiating (12) and using (8) (4) and (5) we have Let now N be a q -slant ruled surface in 3 1 E . By Theorem 3.1 we have 1 2 / k k is constant. Then from (13)  ; .
Let now N be a q -slant ruled surface in 3 1 E . By Theorem 3.1, we have 1 2 / k k is constant. Then from (14) it follows that det( , , ) 0 a a a ′ ′′ ′′′ = .
Conversely, if det( , , ) 0 a a a ′ ′′ ′′′ = , since the curvature 2 k is non-zero from (14) it is obtained that 1 2 / k k is constant and Theorem 3.1 gives that N is a q -slant ruled surface in 3 1 E . .

Theorem 3.4. Non-null ruled surface N is a q -slant ruled surface if and only if
Assume that N is a timelike q -slant ruled surface. From (4) we get Since N is a q -slant ruled surface, 1 2 / k k is constant and by differentiation we have and from (4) Substituting (18) and (19) in (17) gives Using the second equation of (4), (15) is obtained from (20). Conversely, let us assume that (15) holds. Differentiating (19) we obtain and so, Substituting (15) in (22) it follows Now, writing (16) in (23) and using (4) we have On the other hand, from (4) it is obtained Integrating (26) we get that 1 2 / k k is constant and by Theorem 3.1, N is a q -slant ruled surface.
If N is a spacelike ruled surface, then by the similar way it is obtained that N is a qslant ruled surface if and only if (15) holds for  3 1 E In this section, we introduce the definition and characterizations of h -slant non-null ruled surfaces in 3 1 E . First, we give the following definition.  .

h -Slant Ruled Surfaces in
From the second equation of system (31) we have Moreover, (33) Substituting (32) in (33) gives 8 Then from (34) it is obtained that Considering the third equation of system (31), from (35) we have This can be written as Conversely, assume that N is timelike and the function in (29) is constant, i.e., ( ) We define If N is considered as a spacelike ruled surface, then making the similar calculations, it is obtained that N is a h -slant ruled surface if and only if the function ( ) At the following theorem we give a special case for which the first curvature 1 k is equal to 1 and obtain the second curvature for the ruled surface N to be a h -slant ruled surface.
From (43) and (46) we obtain the following differential equation, By integration from (47) we get where c is integration constant. The integration constant can be subsumed thanks to a parameter change s s c → − + . Then (48) can be written as  If the ruled surface N is a spacelike ruled surface then following the same procedure it is easily obtained that N is a h -slant ruled surface if and only if the second curvature is given   From (10) it is clear that a non-null ruled surface N is a -slant ruled surface if and only if it is a q -slant ruled surface. So, all the theorems given in Section 3 also characterize the aslant ruled surfaces.
After these definitions and characterizations of non-null slant ruled surfaces we can give the followings: , , q h a , respectively. If 1 N and 2 N have common central normals i.e., 1 2 h h = at the corresponding points of their striction lines, then 1 N and 2 N are called Bertrand offsets [9].
Similarly, if 1 2 a h = at the corresponding points of their striction lines, then the surface 2 N is called a Mannheim offset of 1 N and the ruled surfaces 1 N and 2 N are called Mannheim offsets [17]. Considering these definitions we come to the following corollaries: