Generalized Timelike Mannheim Curves in Minkowski space-time $E_1^4$

We give a definition of generalized timelike Mannheim curve in Minkowski space-time $E_1^4$. The necessary and sufficient conditions for the generalized timelike Mannheim curve obtain. We show some characterizations of generalized Mannheim curve.


Introduction
The geometry of curves has long captivated the interests of mathematicians, from the ancient Greeks through to the era of Isaac Newton (1647-1727) and the invention of the calculus. It is branch of geometry that deals with smooth curves in the plane and in the space by methods of differential and integral calculus. The theory of curves is the simpler and narrower in scope because a regular curve in a Euclidean space has no intrinsic geometry. One of the most important tools used to analyze curve is the Frenet frame, a moving frame that provides a coordinate system at each point of curve that is "best adopted" to the curve near that point. Every person of classical differential geometry meets early in his course the subject of Bertrand curves, discovered in 1850 by J. Bertrand. A Bertrand curve is a curve such that its principal normals are the principal normals of a second curve. There are many works related with Bertrand curves in the Euclidean space and Minkowski space, [1]- [3]. Another kind of associated curve is called Mannheim curve and Mannheim partner curve. The notion of Mannheim curves was discovered by A. Mannheim in 1878. These curves in Euclidean 3-space are characterized in terms of the curvature and torsion as follows: A space curve is a Mannheim curve if and only if its curvature κ and torsion τ satisfy the relation for some constant β. The articles concerning Mannheim curves are rather few. In [4], a remarkable class of Mannheim curves is studied. General Mannheim curves in the Euclidean 3-space are obtained in [5]- [7] Recently, Mannheim curves are generalized and some characterizations and examples of generalized Mannheim curves are given in Euclidean 4-space E 4 by [8].
In this paper, we study the generalized spacelike Mannheim partner curves in 4−dimensional Minkowski space-time. We will give the necessary and sufficient conditions for the generalized spacelike Mannheim partner curves.

Preliminaries
To meet the requirements in the next sections, the basic elements of the theory of curves in Minkowski space-time E 4 1 are briefly presented in this section. A more complete elementary treatment can be found in [9]. Minkowski space-time E 4 1 is an Euclidean space provided with the standard flat metric given by 1 can locally be spacelike, timelike or null (lightlike) if all of its velocity vectors c ′ (t) are, respectively, spacelike, timelike or null. The norm of v ∈ E 4 1 is given by , c ′ (t) | = 0 for all t ∈ I, then C is a regular curve in E 4 1 . A timelike (spacelike) regular curve C is parameterized by arc-length parameter t which is given by c : I → E 4 1 , then the tangent vector c ′ (t) along C has unit length, that is, c (t) , c (t) = 1 , ( c (t) , c (t) = −1) for all t ∈ I. Hereafter, curves considered are timelike and regular C ∞ curves in E 4 1 . Let T (t) = c ′ (t) for all t ∈ I, then the vector field T (t) is timelike and it is called timelike unit tangent vector field on C.
The timelike curve C is called special timelike Frenet curve if there exist three smooth functions k 1 , k 2 , k 3 on C and smooth non-null frame field {T, N, B 1 , B 2 } along the curve C. Also, the functions k 1 , k 2 and k 3 are called the first, the second and the third curvature function on C, respectively. For the C ∞ special timelike Frenet curve C, the following Frenet formula is Here, due to characters of Frenet vectors of the timelike curve, T, N, B 1 and B 2 are mutually orthogonal vector fields satisfying equations For t ∈ I, the non-null frame field {T, N, B 1 , B 2 } and curvature functions k 1 , k 2 and k 3 are determined as follows where ε is determined by the fact that orthonormal frame field {T (t) , N (t) , B 1 (t) , B 2 (t)}, is of positive orientation. The function k 3 is determined by So the function k 3 never vanishes.
In order to make sure that the curve C is a special timelike Frenet curve, above steps must be checked, from 1 st step to 4 th step, for t ∈ I. Let {T, N, B 1 , B 2 } be the moving Frenet frame along a unit speed timelike curve C in E 4 1 , consisting of the tangent, the principal normal, the first binormal and the second binormal vector field, respectively. Since C is a timelike curve, its Frenet frame contains only non-null vector fields.
3 Generalized timelike Mannheim curves in E 4 1 Mannheim curves are generalized in by [8]. In this paper, we have investigated generalization of timelike Mannheim curves in Minkowski space E 4 1 .
Definition 3.1 A special timelike curve C in E 4 1 is a generalized timelike Mannheim curve if there exists a special timelike Frenet curve C * in E 4 1 such that the first normal line at each point of C is included in the plane generated by the second normal line and the third normal line of C * at the corresponding point under φ. Here φ is a bijection from C to C * . The curve C * is called the generalized timelike Mannheim mate curve of C.
By the definition, a generalized Mannheim mate curve C * is given by the map c * : Here β is a smooth function on I. Generally, the parameter t isn't an arc-length of C * . Let t * be the arc-length of C * defined by If a smooth function f : I → I * is given by f (t) = t * , then for ∀t ∈ I, we have The representation of timelike curve C * with arc-length parameter t * is .
where β is a smooth function on I. Thus, we have Theorem 3.1 If a special timelike Frenet curve C in E 4 1 is a generalized timelike Mannheim curve, then the following relation between the first curvature function k 1 and the second curvature function k 2 holds: where β is a constant number.
Proof Let C be a generalized timelike Mannheim curve and C * be the generalized timelike Mannheim mate curve of C, as following diagram . Thus, the timelike curve C * is reparametrized as follows where β : I ⊂ R → R is a smooth function. By differentiating both sides of equation (3.3) with respect to t, we have On the other hand, since the first normal line at the each point of C is lying in the plane generated by the second normal line and the third normal line of C * at the corresponding points under bijection φ, the vector field N (t) is given by where g and h are some smooth functions on I ⊂ R. If we take into consideration that is, By taking differentiation both sides of the equations (3.5) with respect to t ∈ I, we get (3.6) Since The coefficient of N (t) in equation (3.6) vanishes, that is, Thus, this completes the proof.
Theorem 3.2 In E 4 1 , let C be a special timelike Frenet curve such that its non-constant first and second curvature functions satisfy the equality is a special timelike Frenet curve, then C * is a generalized timelike Mannheim mate curve of C.
Proof The arc-length parameter of C * is given by Under the assumption of , Differentiating the equation c * (f (t)) = c (t) + βN (t) with respect to t the we reach Thus, it is seen that The differentiation of the last equation with respect to t is (3.8) From our assumption, we have Thus, the coefficient of N (t) in the equation (3.8) is zero. It is seen from the equation (3.7), T * (f (t)) is a linear combination of T (t) and B 1 (t) . Additionally, from equation (3.8), N * (f (t)) is given by linear combination of T (t) , B 1 (t) and B 2 (t). On the otherhand, C * is a special timelike Frenet curve that the vector N (t) is given by linear combination of T * (f (t)) and N * (f (t)). Therefore, the first normal line C lies in the plane generated by the second normal line and third normal line of C * at the corresponding points under a bijection φ which is defined by φ (c (t)) = c * (f (t)). This,completes the proof.
Remark 3.1 In 4-diemsional Minkowski space E 4 1 , a special timelike Frenet curve C with curvature functions k 1 and k 2 satisfying k 1 (t) = −β k 2 1 (t) − k 2 2 (t) , it is not clear that a smooth timelike curve C * given by (3.1) is a special Frenet curve. Thus, it is unknown whether the reverse of Theorem 3.1 is true or false. Theorem 3.3 Let C be a special timelike curve in E 4 1 with non-zero third curvature function k 3 . If there exists a timelike special Frenet curve C * in E 4 1 such that the first normal line of C is linearly dependent with the third normal line of C * at the corresponding points c (t) and c * (t), respectively, under a bijection φ : C → C * , iff the curvatures k 1 and k 2 of C are constant functions.
Proof Let C be a timelike Frenet curve in E 4 1 with the Frenet frame field {T, N, B 1 , B 2 } and curvature functions k 1 , k 2 and k 3 . Also, we assume that C * be a timelike special Frenet curve in E 4 1 with the Frenet frame field {T * , N * , B 1 * , B 2 * } and curvature functions k * 1 , k * 2 and k * 3 . Let the first normal line of C be linearly dependent with the third normal line of C * at the corresponding points C and C * , respectively. Then the parameterization of C * is If the arc-length parameter of C * is given t * , then and Moreover, φ : C → C * is a bijection given by φ (c (t)) = c * (f (t)). Differentiating the equation (3.9) with respect to t and using Frenet formulas, we get From last equation, it is easily seen that β is a constant. Hereafter, we can denote β (t) = β, for all t ∈ I. From the equation (3.10), we have Thus, we rewrite the equation (3.11) as follows; The differentiation of the last equation with respect to t is is a non-zero constant number. Thus, from the equation (3.12), we reach where K (t) = k * 1 (f (t)) for all t ∈ I. Differentiating the last equation with respect to t, then we have then we get Arranging the last equation, we find (3.14) Moreover, the differentiation of the equation (3.13) with respect to t is From the above equation, it is seen that Substituting the equations (3.13) and (3.15) into the equation (3.14), we obtain This means that the first curvature function is constant (that is, positive constant). Additionally, from the equation (3.15) it is seen that the second curvature function k 2 is positive constant, too. Conversely, suppose that C is a timelike Frenet curve E 4 1 in with the Frenet frame field {T, N, B 1 , B 2 } and curvature functions k 1 , k 2 and k 3 . The first curvature function k 1 and the second curvature function k 2 of C are of positive constant. Thus, k1 is a positive constant number, say β. The representation of timelike curve C * with arc-length parameter t is (3.16) Let t * denote the arc-length parameter of C * , we have Then, we obtain f ′ (t) = |1 + βk 1 | and that is By differentiating both sides of the above equality with respect to t we find Hence, since k 3 doesn't vanish, we get where ε = sign (k 3 ) denotes the sign of function k 3 . That is, ε is −1 or +1. We can put Differentiating of the last equation with respect to t, we reach and we have Since εk 3 (t) is positive for t ∈ I, we have Thus, we can put Differentiation of the above with respect to t, we get Since f ′ (t) = |1 + βk 1 | and k * 2 (f (t)) N * (f (t)) = k 3 (t) B 2 (t), we have Thus, we obtain B 2 * (f (t)) = δN (t) for t ∈ I, where δ = ∓1. We must determine whether δ is −1 or +1 under the condition that the frame field = 1 for any t ∈ I. Therefore, we get ε = δ. Thus, we get = ε k2 1+βk1 , t ∈ I. By the above facts, C * is a special Frenet curve in E 4 1 and the first normal line at each point of C is the third normal line of C * at corresponding each point under the bijection φ : c → φ (c (t)) = c * (f (t)) ∈ C * . Thus, the proof is completed. The following theorem gives a parametric representation of a generalized timelike Mannheim curves E 4 1 .
Theorem 3.4 Let C be a timelike special curve defined by Here, β is a non-zero constant number, g : U → R and h : U → R are any smooth functions and the positive valued smooth function f : U → R is given by for s ∈ U. Then the curvature functions k 1 and k 2 of C satisfy Proof Let C be a timelike special curve defined by where the prime ( ′ ) denotes the differentiation with respect to t. The unit tangent vector T (t) of the curve C at the each point c (ϕ (t)) is given by By the fact that N (t) = (k 1 (t)) −1 T ′ (t), we get In order to get second curvature function k 2 , we need to calculate k 2 (t) = N ′ (t) − k 1 (t) T (t) . After a long process of calculations and using abbreviations, we obtain  Consequently, from the equations (3.25) and (3.26), we have − P +Q 2 +R 2 +P 2 g 2 + h 2 +R 2 ġ 2 +ḣ 2 +Q 2 g 2 +ḧ 2 − 2PR gġ + hḣ −2RQ ġg +ḣḧ + 2PQ gg + hḧ Substituting the abbreviations into the last equation, we have Substituting the above equation into the equation into the equations (3.27) and (3.28), we obtain The proof is completed.