Timelike tubes with Darboux frame in Minkowski 3-space

In this study, we define a timelike tube around a spacelike curve with timelike binormal by taking Darboux frame instead of Frenet frame in Minkowski 3-space . Subsequently, we compute the Gaussian curvature, the mean curvature, and the second Gaussian curvature of timelike tube with Darboux frame and obtained some characterizations for special curves on this timelike tube around a spacelike curve with timelike binormal.


INTRODUCTION
Tube surface is a special case of the canal surface.A canal surface is defined as one of the parameter family of spheres.Or, a canal surface is the envelope of a moving sphere with varying radius, defined by the trajectory (spine curve) of its centers and a radius function .If the radius function is a constant, then the canal surface is called a tube or tubular surface.Several Geometers have studied canal surfaces and tubes surfaces, and have obtained many interesting results.Maekawa et al (1998) carried out a research on the necessary and sufficient conditions for the regularity of tube surfaces.Also, Ro and Yoon (2009) studied the tubes of Weingarten type in a Euclidean 3-space.Abdel-Aziz and Khalifa (2011) studied the Weingarten timelike tube surfaces around a spacelike curve in Minkowski 3-Space .Recently, Dogan and Yayli (2012) investigated tubes with Darboux frame in Euclidean 3-space.Surface theory had been a popular topic for many researchers in many aspects.Furthermore, using the curves, surfaces, and canal surfaces are the most popular in computer aided geometric design such as designing models of internal and external organs, preparing of terraininfrastructures, constructing of blending surfaces, reconstructing of shape, robotic path planning.
In this study, we investigate the timelike tube surface *Corresponding author.E-mail: skiziltug24@hotmail.com.
around a spacelike curve with timelike binormal by taking the Darboux frame instead of Frenet frame in Minkowski 3-Space and the characterizations of some special curves on this timelike tube are given.

PRELIMINARIES
The Minkowski 3-space is the Euclidean 3-space provided with the indefinite inner product given as: Where,  1 ,  2 ,  3 is natural coordinates of .

Since
is indefinite inner product, recall that a vector can have one of the three causal characters it can be spacelike if and or , timelike if and null (ligtlike) if and .Similarly, an orbitrary curve in can locally called be as spacelike, if its velocity vector is spacelike.Recall that the norm of a vector is given by  = 〈, 〉 and that the spacelike is said to be of unit speed if .Morever, the velocity of curve is the function .
Denote by the moving Frenet frame along the curve in the Minkowski 3-space Then Frenet formula of in the space is defined by (Neill, 1983) Where, and and are curvature and torsion of the spacelike curve , respectively.Then is a spacelike curve with timelike principal normal and spacelike binormal .The vector product of the vectors and is defined by We denote a timelike surface in by: Let be the standard unit of normal vector field on a surface defined by: Where and Then the first fundamental form and the second fundamental form of a surface are defined by(Gray, 1999), respectively Where, On the other hand, the Gaussian curvature and the mean curvature are: , respectively.
If the second fundamental form is non-degenerate; In this case, one define formally the second Gaussian curvature a similar one to Brioschi's formula for the Gaussian curvature obtained on replacing the components of the first fundamental form by those of the second fundamental form as (Khalifa, 2011).

TIMELIKE TUBES WITH DARBOUX FRAME IN
In this section, we define a timelike tube surface around spacelike curve with timelike binormal by taking Darboux frame instead of Frenet frame and compute the coefficients of first and second fundamental form, the Gaussian curvature , the mean curvature , and the second Gaussian curvature for this timelike tube, respectively.
Let be a spacelike unit speed curve with a timelike binormal , where is the arc length parameter of .Consider as a timelike tube surface parametrized by (Khalifa, 2011) Let be timelike surface and be unit speed spacelike curve on the timelike surface .Then, Darboux frame is well-defined along the spacelike curve where is the tangent of and is the unit normal of The derivative formulae of the Darboux frame of is given by: (1) In this formulae and are called the geodesic curvature, the normal curvature and the geodesic torsion, respectively (Uğurlu and Kocayigit, 1996).The relations between geodesic curvature, the normal curvature, the geodesic torsion and are given as follows: Besides, in the differential geometry of surfaces, for a curve lying on a surface the following are well-known:

i) is a geodesic curve if and only if ii) is an asymptotic curve if and only if iii) is a principal line if and only if
Let the center spacelike curve be on the timelike surface Since the characteristic circles of canal surface lie in the plane which is perpendicular to the tangent center of spacelike curve , we can write timelike tube surface with Darboux frame as: (2) (2) where is the unit normal of the surface along the curve Let be a timelike tube surface with Darboux frame in given in Equation 2. So, from the derivative formulas of Darboux frame, partial differentiation of with respect to and are as follows: Therefore, we find the components of the first fundamental form of to be: (3) (3) On the other hand, the unit surface normal vector field is obtained by: (4) (4) The second order partial differentials of are found as: From Equation 4and the last equations we find the second fundamental for coefficients as follows: (5) (5)

Kiziltug and Yayli 33
Thus, the Gaussian curvature , the mean curvature , and the second Gaussian curvature are given by: (6) , respectively.
Where , and are the geodesic curvature, the normal curvature, and the geodesic torsion of , respectively.

SOME CHARACTERIZATIONS FOR SPECIAL CURVES ON THIS TIMELIKE TUBE SURFACE
In this section, we investigate the relation between parameter curves and special curves such as geodesic curves, asymptotic curves, and lines of curvature on this timelike tube surface .

Theorem 1
For the timelike tube surface : i) parameter curves are also geodesic. ii) For parameter curves are also geodesic if and only if , and of satisfy the equation system: (7) (7)

Proof
For -and -parameter curves, we get i) Since , parameter curves are also geodesics.

Proof
Since the center curve is an asymptotic curve, .If we replace in Equation 7, we obtained: .
In the first equation above, if we leave alone and substitute this in the second equation we will get: If we integrate the last equation, it follows that .

Theorem 2
For the timelike tube surface , i) parameter curves cannot also be asymptotic curves on , ii) parameter curves are also asymptotic curve on if and only if is generated by a moving sphere with the radius function.

2
equation above, if we leave alone and substitute this in the second equation we will get: If we integrate the last equation, it follows that .Corollary Let be a asymptotic on timelike tube surface .If parameter curves are also asymptotic on , then the curvatures and of satisfy the equation Where, is a constant.
curves are also asymptotic curve on if and only if