Some Characterizations of Mannheim Partner Curves in Minkowski 3-space

In this paper, we give the characterizations of Mannheim Partner Curves in Minkowski 3-space . Firstly, we classify these curves in . Next, we give some relationships characterizing these curves and we show that Mannheim theorem is not valid for Mannheim partner curves in . Moreover, by considering the spherical indicatrix of the Fr\'enet vectors of those curves, we obtain some new relationships between the curvatures and torsions of the Mannheim partner curves in .


INTRODUCTION
In the differential geometry, special curves have an important role. Especially, the partner curves, i.e., the curves which are related each other at the corresponding points, have drawn attention of many mathematicians so far. The well-known of the partner curves is Bertrand curves which are defined by the property that at the corresponding points of two space curves the principal normal vectors are common. Bertrand partner curves have been studied in ref. [1,2,3,4,13,15]. Ravani and Ku have transported the notion of Bertrand curves to the ruled surfaces and called Bertrand offsets [12]. Recently, Liu and Wang have defined a new curve pair for space curves. They called these new curves as Mannheim partner curves: Let x and 1 x be two curves in the three dimensional Euclidean space 3 E . If there exists a correspondence between the space curves x and 1 x such that, at the corresponding points of the curves, the principal normal lines of x coincides with the binormal lines of 1 x , then x is called a Mannheim curve, and 1 x is called a Mannheim partner curve of x . The pair { } 1 , x x is said to be a Mannheim pair. They showed that the curve 1 1 ( ) x s is the Mannheim partner curve of the curve ( ) x s if and only if the curvature 1 κ and the torsion 1 τ of 1 1 ( ) x s satisfy the following equation Moreover, Oztekin and Ergut [11] studied the null Mannheim curves in the same space. Orbay and Kasap gave [10] new characterizations of Mannheim partner curves in Euclidean 3-space. They also studied [9] the Mannheim offsets of ruled surfaces in Euclidean 3-space. The corresponding characterizations of Mannheim offsets of timelike and spacelike ruled surfaces have been given by Onder and et al [6,7].
In this paper, we give the new characterizations of Mannheim partner curves in Minkowski 3-space 3 1 E . Furthermore, we show that the Mannheim theorem is not valid for Mannheim partner curves in 3 1 E . Moreover, we give some new characterizations of the Mannheim partner curves by considering the spherical indicatrix of some Frénet vectors of the curves. Minkowski 3-space 3   1   E is the real vector space 3  E provided with the standart flat metric  given by   2  2  2  1  2  3 ,

E . Then there is a unique real number
. This number is called the Lorentzian timelike angle between the vectors x and y [6,7].
In this paper, we study the Mannheim partner curves in 3 1 E . We obtain the relationships between the curvatures and torsions of the Mannheim partner curves with respect to each other. Using these relationships, we give Mannheim's theorem for the Mannheim partner curves in Minkowski 3-space 3 1 E .  [5]. The curve * C is timelike. If the curve * C is timelike, then there are two cases. i) The curve C is a spacelike curve with a timelike principal normal. In this case, we say that the pair { } * , C C is a Mannheim pair of the type 1.

MANNHEIM PARTNER CURVES IN MINKOWSKI 3-SPACE
ii) The curve C is a timelike curve. In this case, we say that the pair { } * , C C is Mannheim pair of the type 2.
Case 2. The curve C * is spacelike.
If the curve C * is a spacelike curve, then there are three cases; iii) The curve C * is a spacelike curve with a timelike binormal vector and the curve C is a spacelike curve with a timelike principal normal vector. In this case, we say that the pair { } * , C C is a Mannheim pair of the type 3.
iv) The curve C * is a spacelike curve with a timelike binormal vector and the curve C is a timelike curve. In this case, we say that the pair { } * , C C is a Mannheim pair of the type 4.
v) The curve C * is a spacelike curve with a timelike principal normal vector and the curve C is a spacelike curve with a timelike binormal vector. In this case, we say that the pair { } * , C C is a Mannheim pair of the type 5.
where θ is the angle between the tangent vectors T and * T at the corresponding points of the curves C and * C . From Equations (5) and (6) From the Equations (8) and (9), we obtain * * cosh , sinh ( 1) ds ds ds ds Then by the equations (7) and (10), we see that The proof of the statement given in (ii) can be given by a similar way.  From the statements (iii) and (iv) of the Theorem 3.5, we obtain the following result.  If the pair { } * , C C is a Mannheim pair of the type 2, 3, 4 or 5 we again find that the ration is not constant.

Proposition 3.2. The Mannheim's theorem is invalid for the Mannheim curves in
Then, the pair { } * , C C is a Mannheim pair of the type 3. Figure 1 shows the different appearances of the curves in space in which the curves α * and α are rendered by blue and red colors, respectively.

CONCLUSIONS
In this paper, we give some characterizations of Mannheim Partner Curves in Minkowski 3space 3 1 E . Moreover, we show that the Mannheim theorem is not valid for Mannheim partner curves in 3 1 E . Also, by considering the spherical indicatrix of some Frénet vectors of the Mannheim curves we give some new characterizations for these curves.