The Darboux Trihedrons of Regular Curves on a Regular Time-like Surface

In this work, we study Differential Geometry in the Minkowski 3-space 3 1 R of the curves on a time-like regular surface by using parameter curves which are not perpendicular to each other. The aim of this study is to investigate the formulae between the Darboux Vectors of the time-like curve () c , the time-like parameter curve 1 () c and the space-like parameter curve 2 () c which are not intersecting perpendicularly.


INTRODUCTION
Classical differential geometry of the curves may be surrounded by the topics which are general helices, involute-evolute curve couples, spherical curves and Bertrand curves.Such special curves are investigated and used in some of real world problems like mechanical design or robotics by well-known Frenet-Serret equations.At the beginning of the twentieth century, Einstein's theory opened a door to new geometries such as Minkowski space-time, which is simultaneously the geometry of special relativity and the geometry induced on each fixed tangent space of an arbitrary Lorentzian manifold.In recent years, the theory of degenerate submanifolds has been treated by researchers and some of the classical differential geometry topics have been extended to Lorentzian manifolds.Some authors have aimed to determine Frenet-Serret invariants in higher dimensions.There exists a vast literature on this subject, for instance [1][2][3][4]6,7].In the light of the available literature, in [4] the author extended spherical images of curves to a four-dimensional Lorentzian space and studied such curves in the case where the base curve is a space-like curve according to the signature (+,+,+,-).By using the Darboux vector, various well-known formulas of differential geometry had been produced by [5].Then, in [1], authors had been given these formulae in Minkowski 3-space 3   1 R .
In this work, we investigate the formulae between the Darboux Vectors of the curve () c , the parameter curves 1 () c and 2 () c which are not intersecting perpendicularly.Thus, we will find an opportunity to investigate regular time-like surface by taking the parameter curves which are intersecting under the angle  .
The Darboux Trihedrons of Regular Curves on a Regular Time-Like Surface 173

PRELIMINARIES
To meet the requirements in the next sections, here, the basic elements of the theory of curves in the space 3   1 R are briefly presented (A more complete elementary treatment can be found in [1] ).The Minkowski 3-space 3   1 E provided with the standard flat metric is given by where ( , , ) x x x is a rectangular coordinate system of 3   1 R .Recall that, the norm of an arbitrary vector ,, e e e    we can write their derivative formulaes as follows: w ae be ce Moreover ,  is Lorentzian vectoral product, [1].
Let us take a time-like surface as .For an arbitrary time-like curve   c on time-like surface, the orientation of the Darboux trihedron is written as and the Darboux derivative formulae can be written as follows: ,, And also the Darboux vector of this trihedron is written as where , 1, , E. Özyilmaz

THE DARBOUX VECTOR FOR THE DARBOUX TRIHEDRON OF A TIME-LIKE CURVE
Let us express the parameter curves u=const.as 1 () c and v=const.as .But, these curves intersect under the angle  (not perpendicular).Let any time-like curve that is passing through a point P on the surface be (c).Let us take time-like and space-like parameter curves which are passing through the same point P as , respectively.From   1 , the edges of the Darboux trihedrons of parameter curves are Here, three Darboux trihedrons are written as below: , ss and 2 s be the arc-elements of the curves (c), where , Moreover, because of the parameter curves intersect under the angle  we have 12 t t sh   (3.3), [8].
In the paper the sign will be taken as positive i.e., it will be assumed that Then, the normal vector of time-like surface is is written, [1].
are obtained.From (2.2), we write Proof: From (2.5), we can write the darboux vectors of the ,, Then, if we consider the equations (3.9) , (3.14) , we obtain On the other hand, we have Thus, we have The following geodesic curvature equalities are satisfied for the parameter curves Proof: i) From (3.25) and (3.20), we write Then , from (3.24), if we take inner product both of side is obtained.Similarly, (ii) can be proofed.
According to the theorem 3.  can be written.Thus, we get the theorem.
frame along the time-like curve  

2 (
) c which are on a time-like surface ( , ) y y u v 

1 () c and 2 (
By writing (3.49) in the third expression of (3.40) we obtain N b N w N b w N ds             At the end, if we make equal (3.52) to (3.53) The Darboux Trihedrons of Regular Curves on a Regular Time-Like Surface 175 On the other hand, let us consider the hyperbolic angle between t  .13) If we consider the tangent vectors t 1 and t 2 of the parameter curves (c 1 ) and (c 2 ) on the time-like surface , then we obtain the following relations: