International Journal of Physical Sciences Vol. 8(1), pp. 31-36, 9 January, 2013 Available online at http://www.academicjournals.org/IJPS DOI: 10.5897/IJPS12.602 ISSN 1992 - 1950 ©2013 Academic Journals

Full Length Research Paper

Timelike tubes with Darboux frame in Minkowski 3-space

Sezai Kızıltuğ1* and Yusuf Yaylı2

1Department of Mathematics, Faculty of Science, Atatürk University, Erzurum, Turkey. 2Department of Mathematics, Faculty of Science, Ankara University, Ankara, Turkey.

Accepted 31 December, 2012

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.

Key words: Darboux frame, tube surface, Minkowski space-time, mean and gaussian curvatures, second gaussian curvature, asymptotic and geodesic curves, line of curvature.

INTRODUCTION

Tube surface is a special case of the canal surface. A around a spacelike curve with timelike binormal by taking canal surface is defined as one of the parameter family of the Darboux frame instead of Frenet frame in Minkowski spheres. Or, a canal surface is the envelope of a moving 3-Space and the characterizations of some special sphere with varying radius, defined by the trajectory curves on this timelike tube are given. (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. PRELIMINARIES Several Geometers have studied canal surfaces and tubes surfaces, and have obtained many interesting The Minkowski 3-space is the Euclidean 3-space results. Maekawa et al (1998) carried out a research on provided with the indefinite inner product given as: 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- Where, Space . Recently, Dogan and Yayli (2012) investigated tubes with Darboux frame in Euclidean 3-space. Surface 푦1, 푦2, 푦3 is natural coordinates of . theory had been a popular topic for many researchers in many aspects. Furthermore, using the curves, surfaces, Since is indefinite inner product, recall that a vector and canal surfaces are the most popular in computer can have one of the three causal characters it can aided geometric design such as designing models of be spacelike if and or , timelike if internal and external organs, preparing of terrain- and null (ligtlike) if and . infrastructures, constructing of blending surfaces, Similarly, an orbitrary curve in can locally reconstructing of shape, robotic path planning. called be as spacelike, if its velocity vector is In this study, we investigate the timelike tube surface spacelike. Recall that the norm of a vector is given 훽 = 〈훽, 훽〉 by and that the spacelike is said to be of unit speed if . Morever, the *Corresponding author. E-mail: [email protected]. velocity of curve is the function . 32 Int. J. Phys. Sci.

Denote by the moving Frenet frame respectively. along the curve in the Minkowski 3-space Then If the second fundamental form is non-degenerate; Frenet formula of in the space is defined by In this case, one define formally the second (Neill , 1983) 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).

W here,

and and are curvature and torsion of the spacelike curve , respectively. Then is a spacelike curve TIMELIKE TUBES WITH DARBOUX FRAME IN with timelike principal normal and spacelike binormal . The vector product of the vectors and In this section, we define a timelike tube surface around is defined by 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.

We denote a timelike surface in by: 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 the standard unit of normal vector field on a surface defined by:

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 Where and Then the first and is the unit normal of The derivative formulae of fundamental form and the second fundamental form the Darboux frame of is given by: of a surface are defined by(Gray, 1999), respectively

(1)

Where,

In this formulae and are called the geodesic curvature, the normal curvature and the geodesic torsion, On the other hand, the Gaussian curvature and the respectively (Uğurlu and Kocayigit, 1996). The relations mean curvature are: 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 Kiziltug and Yayli 33

well-known: Thus, the Gaussian curvature , the mean curvature , and the second Gaussian curvature are given by: 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 (6) surface lie in the plane which is perpendicular to the (6) tangent center of spacelike curve , we can write timelike tube surface with Darboux frame as:

, (2) (2) respectively.

Where , and are the geodesic curvature, the where is the unit normal of the surface along the normal curvature, and the geodesic torsion of , curve respectively. 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 SOME CHARACTERIZATIONS FOR SPECIAL to and are as follows: 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 . Therefore, we find the components of the first fundamental form of to be: Theorem 1

(3) For the timelike tube surface : (3) i) parameter curves are also geodesic. On the other hand, the unit surface normal vector field is obtained by : Forii) parameter curves are also geodesic if and only if , and of satisfy the equation system: (4) (4)

The second order partial differentials of are found as: (7) (7)

Proof

For - and - parameter curves, we get

From Equation 4 and the last equations we find the second fundamental for coefficients as follows:

(5) i) Since , parameter curves are also geodesics. (5) 34 Int. J. Phys. Sci.

ii) Because , and are linearly independent, if and only if parameter curves are also asymptotic on , then the curvatures and of satisfy the equation

Where, is a constant.

Proof By the last two equations, we have Since the center curve is an asymptotic curve, . If we replace in Equation 7, we obtained:

Then , and hold the equation system

.

In the first equation above, if we leave alone and This completes proof. substitute this in the second equation we will get:

Corollary 1

Let be a geodesic on timelike tube surface . If If we integrate the last equation, it follows that parameter curves are also geodesic on , then the curvatures and of satisfy the equation: .

Theorem 2

Where, is a constant. For the timelike tube surface ,

i) parameter curves cannot also be asymptotic curves Proof on , ii) parameter curves are also asymptotic curve on Since the center curve is an geodesic curve, . if and only if is generated by a moving If we replace in Equaion 7, we obtain: sphere with the radius function.

푘 cosh 푣+푘 sinh 푣 푔 푛 푟 = 2 2 = c (푘푔cosh 푣+푘푛 sinh 푣) +푡푟 (8) . such that is a constant.

In the first equation above, if we leave alone and substitute this in the second equation we will get: Proof

2 2 i) As 〈푍 × 푀푣푣 〉 = −푟(cosh푣) + 푟 sinh푣 = 푟 ≠ 0 푣 − I f we integrate the last equation, it follows that parameter curves cannot also be asymptotic curves on . . ii) parameter curves are also asymptotic curve on if and only if

Corollary 2

Let be a asymptotic on timelike tube surface . If From this, we get the radius function: Kiziltug and Yayli 35

Theorem 3

The parameter curves of are also lines of curvature if and only if the center curve is a line of This completes proof. curvature on

Corollary 3 Proof Let parameter curves are also asymptotic curves on . From Equations 3 and 5 we have: i) If the center curve is a geodesic curve on , then

푘sinh푣 푟 = = c. According to theorem of line of curvature, the parameter 푘2 sinh푣 2 + 휏2 curves on a surface are also lines of curvature if and only if . From , it concludes that , ii) If the center curve is a asymptotic curve on that is, is a line of curvature on . , then:

Theorem 4

For the timelike tube surface , iii) If the center curve is a line of curvature on i) İf the center curve is a geodesic curve on , , then: then the Gaussian curvature , the mean curvature , and the second Gaussian curvature of surface are as follows:

Proof

Because parameter curves are also asymptotic curves on , from Equation 8:

i) Since is a geodesic curve on , So, ii) İf the center curve is a asymptotic curve on , and . If we replace these in Equation 8, we , then the Gaussian curvature , the mean curvature and the second Gaussian curvature of surface are as follows: obtained: ii) Since is a asymptotic curve on , So, and . If we replace these in Equation 8),

we obtained: iii) Since is a line of curvature on , . If we replace these in Equation 8, we will get:

36 Int. J. Phys. Sci.

Proof ACKNOWLEDGEMENT i) Because is a geodesic curve on , The authors would like to thank the referee for his/her So, and . If we replace these in Equation 6, valuable suggestions which improved the first version of we obtained: the paper.

REFERENCES

Abdel AHS, Khalifa SM (2011). Weingarten Timelike Tube Surfaces around a Spacelike Curve. Int. J. Math. Anal. 5(25):1225-1236. Doğan F, Yayli Y (2012). Tubes with Darboux Frame. Int. J. Contemp. Math. Sci. 7(16):751-758. Gray A (1999). Modern Differential Geometry of Curves and Surfaces with Mathematica. second ed, CrcPress, USA. Maekawa T, Patrikalakis MN, Sakkalis T, Yu G (1998). Analysis and Applications of Pipe Surfaces. Comput. Aided Geom. Des. 15:437- 458. O' Neill B (1983). Semi-Riemannian geometry with applications to relativity. Academic press, New York. ii) Because is a asymptotic curve on , Ro JS, Yoon DW (2009). Tubes of Weingarten Types in a Euclidean 3- space. J. Chngchng. Math. Sci. 3(22):359-366. So, and . If we replace these in Equation 6, Uğurlu HH, Kocayigit H (1996). The Frenet and Darboux Instantaneous we obtain: Rotain Vectors of Curves on Time-Like Surfaces. Math. Compt. Appl. 1(2):133-141.