ON TIME-LIKE RULED SURFACES IN MINKOWSKI 3-SPACE

In this paper, a study of time-like ruled surfaces in Minkowski 3-space is investigated by strictly connected time-like straight line moving with Darboux’s frame along a differentiable space-like curve. By using the striction curve and the distribution parameter of time-like ruled surfaces, some theorems related to the geodesic curvature and the second fundamental form tensor are obtained.


Introduction
Let R 3 be endowed with Lorentzian inner product of X and Y is defined by ⟨X, Y ⟩ = x 1 y 1 + x 2 y 2 − x 3 y 3 , for X, Y ∈ R 3 . (R 3 , ⟨, ⟩) is called Minkowski 3-space denoted by R 3 1 . Then Lorentzian vector product of X and Y is defined by X ∧ Y = (x 2 y 3 − x 3 y 2 , x 3 y 1 − x 1 y 3 , x 2 y 1 − x 1 y 2 ), for any vectors X, Y ∈ R 3 1 , (see [8]). The vectors X ∈ R 3 1 are called a space-like, time-like and null (light-like) vector if ⟨X, X⟩ > 0 or X = 0, ⟨X, X⟩ < 0 and ⟨X, X⟩ = 0 for all X ̸ = 0, respectively. If ⟨X, Y ⟩ = 0 for all X and Y , the vectors X and Y are called orthogonal in the sense of Lorentzian. For X ∈ R 3 1 , the norm of X is defined by ∥X∥ = √ |⟨X, X⟩| and X is said to be a unit vector if ∥X∥ = 1, (see [8]).
Let α(t) : I ⊂ R → R 3 1 , be a smooth regular curve in R 3 1 . For any t ∈ I, the curve α is said to be a space-like, time-like and null curve if the velocity vector α ′ (t) is a space-like, time-like or null vector, respectively, (see [8]).
A surface in Minkowski 3-space is called a time-like surface if the induced metric on the surface is a Lorentzian metric, i.e., the normal on the surface is a space-like vector. A time-like ruled surface in R 3 1 is obtained by a space-like straight line moving along a time-like curve or by a time-like straight line moving along a space-like curve. A time-like ruled surface is given by the parametrization φ : I × R→R 3 1 , φ(t, υ) = α(t) + υX(t), in R 3 1 , where the curve α is called the base curve and X is called the ruling, (see [7]). Let α = α(s) be a unit speed space-like curve in R 3 1 ; The natural curvature and torsion of α(s) are denoted by k 1 (s), k 2 (s), respectively. Consider the Frenet frame {V 1 , V 2 , V 3 } attached to the space-like curve α = α(s) such that V 2 = V 2 (s) is any principal normal vector field of type time-like, V 3 = V 3 (s) is any binormal vector field of type space-like and V 1 = V 1 (s) is the unit tangent vector field. Then the derivative formulas of Frenet frame {V 1 , V 2 , V 3 } can be written as  We also define Darboux frame attached to the space-like curve α = α(s) such that V 1 = V 1 (s) is a unit tangent vector field, N = N (s) is a unit vector field of type space-like and G = G(s) is a binormal vector field of type time-like defined by where k g , k n and τ g are defined by and φ is any hyperbolic angle between the time-like vectors V 2 and G, and called as geodesic curvature, normal curvature and geodesic torsion, respectively, (see [9]).
In this paper, we study some characteristic properties of time-like ruled surfaces related to the geodesic curvature and the second fundamental form tensor by the means of the information given above.

Methodology
Let α be a space-like curve with parameterized by arc length on a time-like ruled surface. In this situation, the derivative formulas of Darboux frame {V 1 , G, N } at the point α(s) can be written as where k g , Π be the geodesic curvature and the second fundamental form tensor, respectively. Now, we compute the distribution parameter of the time-like ruled surface M . Let P X be a distribution parameter of time-like ruling X. Then we have where X ′ is a time-like vector field.
Analogously, P X be a distribution parameter of space-like ruling X. So we get where X ′ is a space-like vector field.
Also, we obtain the striction curve of any time-like ruled surface M . If the distance between the base curve and the striction point of the time-like ruled surface M is u, then the position vector α can be expressed in the form The parameter u can be expressed in terms of the position vector of the space-like directrix. So we get Thus the striction curve is given by Conversely, if ⟨X ′ , X ′ ⟩ = 0, then the time-like ruled surface M is a cylinder. For time-like ruled surfaces, the striction curve can be taken as directrix. Hence, we must take u = 0 in Eq. (4). Now, we study neighborhood of a point P in the time-like ruled surface M . We assume that time-like ruling X is orthogonal to the directrix α at each point. Thus we can write ⟨T, X⟩ = 0, where T is a tangent vector field of directrix α. Let N be a local unit normal for this coordinate neighborhood. So the unit vector fields T, X and N give orthonormal frame along the space-like curve α and we obtain Frenet frame for this system. Similarly, we can obtain Darboux frame {T, X * , N * } at the same point. We can get the following derivative formulas of these frames T, X and N, Taking υ as constant, we get a space-like curve φ υ = α ′ (s) + υX ′ (s) on M with tangent Hence, we can say that the space-like vector field A is normal to X. Methodologically, we investigate on a time-like ruled surfaces in two different ways. Firstly, we examine on a time-like ruled surfaces with time-like rulings. Then similiar calculations are performed using time-like directrix.

Results
In this section, we firstly study on a time-like ruled surface swept out by a time-like straight line X moving along a differentiable space-like curve α.
Let us consider M as a time-like ruled surface in R 3 1 . There always exists Frenet and Darboux frames at each point of a time-like curve α on this surface. If we coincide these frames at the point P , we get Eq. (7). Thus the following theorems can be given Proof. Since the tangent space along a ruling is uniquely spanned by the system {A, X}, the space-like vector field N is constant along a space-like ruling. Thus the system {A, T } is linearly dependent. Let υ be a constant. Then we get a space-like curve where A is a space-like vector field and N ̸ = 0. By the meaning of Eq. (8), we obtain Π(T, X) = 0, which completes the proof.

Theorem 3.2. Let M be a time-like ruled surface in R 3 1 . The time-like ruled surface M is developable if and only if the distribution parameter of the time-like ruled surface is zero.
Proof. With the help of Eq. (2) and Theorem 3.1, we find P X = 0.
Proof. We consider two rulings passing at the points α(s 0 ) and α(s 1 ), where s 0 < s 1 . The distance between these rulings along an orthogonal trajectory is given by where A is a space-like vector field. Using Eq. (8), we get To estimate the minimizing the value of J(υ), we use the derivative function of J(υ). So we conclude that completing the proof. Proof. Let any point φ(s, υ 0 ) be a striction point on the main ruling passing at the point α(s). We must show that ⟨D T X, A⟩ = 0. Since ⟨X, X⟩ = −1, we obtain ⟨D T X, X⟩ = 0. Also, since A = φ υ and X be the tangent vector,then we have the following equation .
Conversely, the tangent vector field of the time-like curve φ υ0 : I → M is given Since D T X is normal to the tangent plane, we get Therefore, the point φ(s, υ 0 ) is a striction point on a main ruling. Proof. Let X be a time-like vector field of which the rulings pass through the point P ∈ M . An orthogonal basis {X, Y } can be obtained, where Y is a space-like vector field. Let S be the matrix corresponding to the shape operator of M derived from N . Also, the Gaussian curvature function is defined as Differentiating Eq. (11) with respect to υ, we get So the value of υ = − k g k 2 g + [⟨Π(T, X), N ⟩] 2 gives the maximum of K(s, υ) since Hence, Gaussian curvature function has the maximum value at the striction point on each one of the rulings since the striction point corresponds to the value Then using Eq. (10), we find On the other hand, combining Eqs. (2) and (12), we obtain since k g = 0 at the striction point. Therefore, we have The above relation means that the maximum value of the Gaussian curvature function of the time-like ruled surface M along a ruling only depends on the distribution parameter. The value of K max (υ) is unique along a ruling which completes the proof.
The important theorem concerning with the striction point of any ruled surfaces in Euclidean 3-space was given by Chasles (see [1], [3] ). Now, we can give this theorem for time-like ruled surfaces in Minkowski 3-space related to the second fundamental form tensor as follows:

Theorem 3.6. Let M be a non developable time-like ruled surface in R 3 1 and θ be the angle between the normal vector at the point of a ruling and the normal vector at the striction point of this ruling. Then the tan θ is proportional to the distance between these points and the coefficient of proportionality is the inverse of the distribution parameter.
Proof. Let the point φ(s, 0) be a striction point of M for υ = 0. The curve of α : I → M is an orthogonal trajectory which passes through the striction point φ(s, 0). Then D T X is normal vector at υ = 0 and k g = 0. Thus the distribution parameter P X is written as follows: Also, the normal vector N v along the ruling is given by On the other hand, N and N υ are the unit space-like vectors. Let θ be the angle between N and N υ . Therefore, we obtain Thus we have By the means of Eq. (2), we obtain This is the desired result. Now, we study on a time-like ruled surface swept out by a space-like straight line X moving along a differentiable time-like curve α.
We consider M is a time-like ruled surface in R 3 1 . There exists Frenet and Darboux frames at each point of a time-like curve α on this surface. If we coincide these frames at the point P , we obtain Thus the following theorems can be given: Proof. Since the tangent space along a ruling is spanned by the system {A, X}. So this tangent space is unique along a ruling. Then the space-like vector field N is constant along a time-like ruling α. Hence, the system {A, T } is linearly dependent. Let υ be a constant, then we get a time-like curve φ υ = A = α ′ (s) + υX ′ (s) on M with tangent where A is a time-like vector field. By using Eq. (14), we obtain Π(T, X) = 0.  Proof. We consider two rulings passing at the points α(s 0 ) and α(s 1 ), where s 0 < s 1 . The distance between these rulings along an orthogonal trajectory is given by where A is a time-like vector field. Then using Eq. (14), we have To find the value of υ, which maximizes J(υ), we use the derivative function of J(υ). So we deduce which completes the proof. Proof. Let φ(s, υ 0 ) be a striction point on the main ruling passing at the point α(s). We must show that ⟨D T X, A⟩ = 0. If we take the derivative of equation ⟨X, X⟩ = 1, then we obtain ⟨D T X, X⟩ = 0. By direct computation, the value ⟨D T X, A⟩ is obtained as follows: ] .
Since φ(s, υ 0 ) is a striction point, we have Then substituting the value (15), we obtain ⟨D T X, A⟩ = 0. It means that Conversely, the tangent vector field of the time-like curve φ υ0 : I → M is given Since D T X is normal to the tangent plane, we also have Therefore, the point φ(s, υ 0 ) is a striction point on a main ruling which shows desired result. Proof. Let X be a space-like vector field of which the rulings through the point P ∈ M . An orthogonal basis {X, Y } can be obtained, where Y is a time-like vector field. Let S be the matrix corresponding to the shape operator of M derived from N . Then Gaussian curvature function is given by Differentiating Eq. (17) with respect to υ, we get Thus the value of υ = k g gives the minimum of K(s, υ), since Hence, Gaussian curvature function has the minimum value at the striction point on each one of the rulings, since a striction point corresponds to the value With the help of Eq. (17), we have On the other hand, combining Eqs. (3) and (18), we obtain since k g = 0 at the striction point. Therefore, we find which means the minimum value of Gaussian curvature function of time-like ruled surface M along a ruling only depends on the distribution parameter. The value of K min (υ) is unique along a ruling.  φ(s, 0). Then D T X is normal vector at υ = 0 and k g = 0. Thus the distribution parameter P X is written as follows: Also, the normal vector N v along the ruling is given by On the other hand, N and N υ are the unit space-like vectors. Let θ be the angle between N and N υ . Therefore, we obtain Thus we get By using Eq. (3), we obtain Therefore, we have υ = tan θP X , which completes the proof.

Discussion
The theory of special relativity was created by Einstein in 1905. Then Poincare has shown that the transformation of this special relativity (The Lorentzian Transformation) is invariant of rotations in 3-dimensional Euclidean space. Following these studies, Minkowski investigated this analogy and he realized that space and time could be linked in 4-dimensional continuum, with coordinates (x, y, z, ict) where c is the speed of light in a vacuum and the variable is i = √ −1, called "pure imaginary". The Lorentzian transformations are no more than rotations in Minkowski space. The constancy of the speed of light can be understood by considering the emission of a light signal at any position and the reception of the same signal in a differerent position. Using comparing these two events, the equation of the speed of light is obtained and it can be interpreted as the expressing invariance of a null distance in a space with three real dimensions and one imaginary dimension.
The space-time continuum provided a framework for all works in relativity. Einstein was developed a frame in general relativity. However, the simplier Minkowski space was not able to deal with accelerating reference frames and was to be replaced by the more versatile Riemannian space.
The purpose of this paper is to give an informal introduction to the geometry of 3-dimensinonal Minkowski space in the light of the parallelism with 3-dimensional Euclidean space. The well-known theorems in 3-dimensional Euclidean space are proved for a time-like ruled surface with time-like rulings or time-like directrix.

Conclusion
As revealed from the foregoing relations and theorems in section 3, we have the followings: