On the Involute-Evolute of the Pseudonull Curve in Minkowski 3-Space

We have generalized the involute and evolute curves of the pseudonull curves α in E3 1 ; that is, α is a spacelike curve with a null principal normal. Firstly, we have shown that there is no involute of the pseudonull curves α in E3 1 . Secondly, we have found relationships between the evolute curve β and the pseudonull curve α in E3 1 . Finally, some examples concerning these relations are given.


Introduction
The general theory of curves in an Euclidean space (or more generally in a Riemannian manifold) was developed a long time ago, so now, we have a deep knowledge of its local geometry as well as its global geometry.In the theory of curves in Euclidean space, one of the important and interesting problems is the characterizations of a regular curve.And, in particular, the involute-evolute of a given curve is a well-known concept in the classical differential geometry (see [1]).
At the beginning of the twentieth century, A. Einstein's theory opened a door for use of new geometries.One of them is simultaneously the geometry of special relativity and the geometry induced on each fixed tangent space of an arbitrary Lorentzian manifold and was introduced and some of classical differential geometry topics have been treated by the researchers.According to reference (see [2,3]), the involute and evolute curves of the spacelike curve  in E 3 1 with a spacelike binormal or a spacelike principal normal in Minkowski 3-space have been investigated.Also, Bükcü and Karacan studied the involute and evolute curves of the timelike curve in Minkowski 3-space [4].
In this paper, we have generalized the involute and evolute curves of the pseudonull curves  in E 3  1 ; that is,  is a spacelike curve with a null principal normal.Firstly, we have shown that there is no involute of the pseudonull curves  in E 3 1 .Secondly, we have found relationships between the evolute curve  and the pseudonull curve  in E 3  1 .Finally, some examples concerning these relations are given.

Preliminaries
The Minkowski 3-space E 3  1 is the Euclidean 3-space E 3 provided with the standard flat metric given by where ( 1 ,  2 ,  3 ) is a rectangular coordinate system of E 3 1 .Since  is an indefinite metric, recall that a vector V ∈ E 3 1 can have one of three Lorentzian causal characters: it can be spacelike if (V, V) > 0 or V = 0, timelike if (V, V) < 0, and null (lightlike) if (V, V) = 0 and V ̸ = 0.In particular, the norm (length) of a nonnull vector V is given ‖V‖ = √|(V, V)|, and two vectors V and  are said to be orthogonal, if (V, ) = 0.A lightlike vector  is said to be positive (resp., negative) if and only if  1 > 0 (resp.,  1 < 0) and a timelike vector  is said to be positive (resp., negative) if and only if  1 > 0 (resp.,  1 < 0).

Journal of Applied Mathematics
Next, recall that an arbitrary curve  = () in E 3  1 can locally be spacelike, timelike, or null (lightlike), if all of its velocity vectors   () are, respectively, spacelike, timelike, or null (lightlike).If (  (),   ()) = ±1, the nonnull curve  is said to be of unit speed (or parameterized by arc length function ).
We denote by { ⃗ , ⃗ , ⃗ } the moving Frenet frame along the curve ().Then ⃗ , ⃗ , and ⃗  are the tangent, the principal normal, and the binormal vector of the curve , respectively.Depending on the causal character of the curve , we have the following Frenet-Serret formulas.
If  is a null space curve with a spacelike principal normal ⃗ , then the following Frenet formulas hold where For a null curve,  can take only two values:  = 0 when  is a null straight line or  = 1 in all other cases.If  is a pseudonull curve, that is,  is a spacelike curve with a null principal normal , then the following Frenet formulas hold: where For a pseudonull curve,  can take only two values:  = 0 when  is a straight line or  = 1 in all other cases.If  is a spacelike space curve with a spacelike principal normal ⃗ , then the following Frenet formulas hold where If  is a spacelike space curve with a timelike principal normal ⃗ , then the following Frenet formulas hold where If  is a timelike space curve, then the following Frenet formulas hold where see [5].
If the curve () is nonunit speed, then If the curve () is unit speed, then see [6].
In this study we are going to have the  curve as nonline pseudonull curve or  = 1.

The Involute of the Pseudonull Curve
Definition 1.Let a pseudonull curve  :  → E 3  1 and a curve  :  → E 3 1 be given.For all  ∈ , then the curve  is called the involute of the curve , if the tangent vector of the curve  at the point () passes through the tangent at the point () of the curve  and where { ⃗ , ⃗ , ⃗ } and { ⃗  * , ⃗  * , ⃗  * } are Frenet frames of  and , respectively.This definition suffices to define this curve mate as (see Figure 1).
Proof.Let  :  → E 3 1 be involute of  in E 3 1 .We assume that  is distinct from .Then we can write where  is a  ∞ -function on .Differentiating (17) and by using Frenet formulas given in (4), we get If we take the inner product with ⃗ () on both sides of (18), we have Recalling definition of the involute curve couple, ( ⃗ (), ⃗  * ()) = 0 and, by using (5), we get for all  ∈ .Thus, since  ∈ R − {0}, we get () =  − ; that is, () is a nonzero constant.In this case, we can write (17) as follows: and from (18) we get Here we notice that and by using (5) we get Thus, since ( ⃗  * (), ⃗  * ()) = 0, we get that ⃗  * is a null vector; that is, curve  is a null curve.
Differentiating (24) with respect to  and by using Frenet formulas given in (4), we have and by using (5) we get Since  * () ̸ = 0 for all  ∈ , we get Thus, we obtain that ⃗  * is a null vector.This is a contradiction with null curve .Then, there is no involute of the curve  in Minkowski 3-space E 3  1 .

The Evolute of the Pseudonull Curve
Definition 3. Let the pseudonull curve  and a curve  with the same interval be given.For all  ∈ , then the curve  is called the evolute of the curve , if the tangent vector of the curve  at the point (s) passes through the tangent at the point () of the curve  and  (34) Recalling definition of the evolute curve couple, ( ⃗ (), ⃗  * ()) = 0, and by using (5), we get for all  ∈ .In this case, we can write (31) as follows: where  is a  ∞ -function on .Since this line passes through the point (), the vector () − () is perpendicular to the vector ⃗ ().From ( 36) and (37), the vector field   () = ⃗  * () is parallel to the vector field () − ().Then, we have 4.1.Special Cases.If () = 0 for all  ∈ , then (39) is not defined.But, from (36) and (37), we get Here we notice that and by using ( 5) we get Thus, since ( ⃗  * (), ⃗  * ()) = 0, we get that ⃗  * is a null vector; that is, the curve  is a null curve.Differentiating (41) and by using Frenet formulas given in (4), we have and from (5) we get Since the space curve  is a null curve, then from ( ⃗  * (), ⃗  * ()) = 1 and  * () = 1 we get Case 1.If () = −1 from (46), then we can write (41) and (44) as follows: Differentiating (48) and by using Frenet formulas given in ( 2) and ( 4), we get By using Frenet formulas given in (2), we get Conclusion 1.Let a space curve  be the evolute of the pseudo null curve  in E 3 1 and let  * and  * are the curvatures of the curve .If the torsion  of the curve  is for all  ∈ , then  and  are helices and the curvatures of the curve  are for all  ∈ .
Case 2. If () = 1 from (46), then we can write (41) and (44) as follows: Differentiating (54) and by using Frenet formulas given in ( 2) and (4), we get By using Frenet formulas given in (2), we get Conclusion 2. Let a space curve  be the evolute of the pseudo null curve  in E 3 1 and let  * and  * are the curvatures of the curve .If the torsion  of the curve  is for all  ∈ , then  and  are helices and the curvatures of the curve  are for all  ∈ .
From ( 4), (5), and (37), we get and by using (60) we get Case 3. If  is a unit spacelike curve, then from (62) we have Solving the differential equation ( 63), we get where  1 and  2 are constant.Then, from (36) we can write Case 4. If  is a unit timelike curve, then from (62) we have Solving the differential equation (65), we get where  1 and  2 are constant.Then, from (36) we can write Case 5.If  is a null curve, then from (62) we have Solving of the differential equation (65), we get where  is constant.This is a contraction with () ̸ =  ( = constant).Then, the curve  is not defined.