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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 810640, 5 pages
The Ruled Surfaces According to Bishop Frame in Minkowski 3-Space
Department of Mathematics, Erciyes University, 38039 Kayseri, Turkey
Received 8 April 2013; Accepted 18 June 2013
Academic Editor: Chuanzhi Bai
Copyright © 2013 Nural Yüksel. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We investigate the ruled surfaces generated by a straight line in Bishop frame moving along a spacelike curve in Minkowski 3-space. We obtain the distribution parameters, mean curvatures. We give some results and theorems related to be developable and minimal of them. Furthermore, we show that, if the base curve of the ruled surface is also an asymtotic curve and striction line, then the ruled surface is developable.
Recently, the theory of surfaces and their transformations has been studied extensively in differential geometry. The ruled surfaces have been a powerful subject in the Minkowski space for line geometry for a long time. In the literature, Kobayashi  was the first author to address this problem and examined minimal spacelike ruled surfaces in the Minkowski . Kim and Yoon  have classified the Lorentz surfaces.
Izumiya and Takeuchi  obtained some characterizations for ruled surfaces. Turgut and Hacısalihoğlu [4, 5] defined spacelike ruled surfaces and obtained some characterizations in the three-dimensional Minkowski space. Yaylı  obtained the distribution parameter of a spacelike ruled surface generated by a spacelike straight line in Frenet frame along a spacelike curve. Yaylı and Saracoglu [7, 8] studied timelike and spacelike developable ruled surfaces in Minkowski space. Orbay and Aydemir  obtained the distrubition parameter, mean curvature, and Gaussian curvature, and some new results and theorems were given for developable and minimal spacelike ruled surfaces.
In this paper, making use of the method in a paper of Yaylı , we obtained some characterizations for spacelike Ruled surfaces according to Bishop frame in Minkowski 3-space.
Let be a Minkowski 3-space with the metric tensor . The norm of is defined by . A vector is said to be spacelike if or , timelike if , and lightlike (or null) if and .
Let be a smooth regular curve in . We say that is a spacelike (resp. timelike, lightlike) if , a spacelike (resp. timelike, lightlike) vector for all .
A surface in the Minkowski 3-space is called a spacelike surface if the Lorentz metric on the surface is a positive definite . A ruled surface is a surface swept out by a straight line moving along a curve . The various positions of the generating line are called the rullings of the surface. Such a surface has a parametrization in the ruled form as follows: where is the base curve and is the director vector along . If the tangent plane is constant along a fixed rulling, then the ruled surface is called a developable surface. The remaining ruled surfaces are called skew surfaces . The spacelike ruled surface in is given by the parametrization where is a differentiable spacelike curve parametrized by its arc length in and is the director vector of the director curve such that is ortogonal the tangent vector field of the base curve .
Denote by the moving Frenet frame along the regular curve with arc-lenght parameter . The Frenet trihedron consists of the tangent vector , the principal normal vector , and the binormal vector . If is a spacelike curve with a spacelike binormal, then the Frenet frame has the following properties: where
The Bishop frame or parallel transport frame is an alternative approach to defining a moving frame that is well defined even when the curve has a vanishing second derivative. One can express parallel transport of an orthonormal frame along a curve simply by parallel transporting each component of the frame. The tangent vector and any convenient arbitrary basis for the remainder of the frame are used.
Let us consider the Bishop frame of the spacelike curve such that the spacelike unit tangent vector, is timelike unit normal vector, and the spacelike unit binormal vector. So scalar product and cross product of the vectors are given by The Bishop frame is expressed as One can show that so that and effectively correspond to a cartesian coordinate system for the polar coordinates , with .
Remark 1. From the definition of the argtanh function we assume that .
The distribution parameter, the mean curvature, and the Gaussian curvature of the ruled surface are given by where is the Levi-Civita connection on .
Theorem 2. A spacelike ruled surface is a developable surface if and only if the distrubition parameter of the spacelike ruled surface is zero .
The foot on the main rulling of the common perpendicular of two constructive rullings in the ruled surface is called a central point. The locus of the central point is called the striction curve. The parametrization of the striction curve on the ruled surface is given by
3. One Parameter Spatial Motion in
Let be a spacelike curve and be its Bishop frame where , and are the tangent, principal normal, and binormal vectors of the curve , respectively. and are spacelike vectors and is a timelike vector.
The two coordinate systems and are orthogonal coordinate systems in which represent the moving space and the fixed space , respectively.
Let be a unit spacelike vector such that
We can obtain the distrubition parameter of the spacelike ruled surface generated by a straight line of the moving space . Differentiating (13) with respect to , we get By using the Bishop frame in (15), we obtain From (10) we get
Theorem 3. Let be a spacelike ruled surface given by the parametrization (2). is developable if and only if either the director vector lies in the plane generated by and or the base curve is a planar curve such that the curvatures of , and satisfy
4. Special Cases
Let be a spacelike ruled surface given by the parametrization (2), and, be the director vector of the base curve .
4.1. The Case (Spacelike)
In this case, , thus from (17) Hence the following theorem is hold.
Theorem 4. During the one-parameter spatial motion the spacelike ruled surface in the fixed space generated by the tangent line of the curve in the moving space is developable.
4.2. The Case (Spacelike)
From Theorem 3, it is obvious that .
4.3. The Case
In this case, is zero. So, the director vector is given by The distribution parameter of the ruled surface is given by The ruled surface is developable if and only if . Thus
If , this is case 4.1. If the second curvature is zero, then we can say that the base curve is a planar curve.
4.4. The Case
In this case, is zero. So, the director vector is given by
From (17) the distribution parameter is obtained as if and only if If is zero, this is case 4.3. If is zero, this is The case 4.1.
From Theorem 3, is a developable spacelike ruled surface.
4.5. The Case
From Theorem 3 it is obvious that the spacelike ruled surface is developable.
By using (11) we compute the mean curvatures of the spacelike ruled surfaces generated by spacelike vectors , and .
Corollary 6. The surface is minimal if and only if is a planar curve.
Proof. Let be minimal. In this case, from (31), we get
Conversely, let be a planar curve. Then implies that . This completes the proof.
Proposition 7. Let be a spacelike ruled surface generated by the binormal line of the base curve . From (11) the mean curvature is obtained as follows: So, the following result may be given.
Corollary 8. According to the Bishop frame, there is no minimal spacelike ruled surface generated by the binormal line in .
Proposition 9. Let be a spacelike ruled surface which is given by the parametrization (2): where where is a unit normal vector of the spacelike ruled surface .
Proposition 10. Let be a spacelike ruled surface given by the parametrization (2). If the base curve of is also a striction curve, then the curvature functions and of the base curve satisfy the following equation:
Proof. Let the base curve be the striction curve. Thus, from (12), Then we have
Hence the following result holds.
Corollary 11. Let be a spacelike ruled surface given by the parametrization (2). If the base curve of is also striction curve, then is a planar curve.
Proposition 12. Let be a spacelike ruled surface given by the parametrization (2). If the base curve of is also asymtotic curve, then
Proof. We assume that the base curve of the surface is the asymtotic curve. In that case, From (46), we have
Theorem 13. Let the base curve of the surface be an asymtotic curve. If the base curve of is also a striction curve, the spacelike ruled surface is developable.
Proposition 14. Let be a spacelike ruled surface given by the parametrization (2). We obtain the following results for the spacelike ruled surfaces.(i) The -parameter curve of is also an asymtotic curve if and only if (ii) The -parameter curve of is also an asymtotic curve.
Proof. (i) If the -parameter curve of is also an asymtotic curve, then From (36), we obtain (46).(ii) If the -parameter curve of is also an asymtotic curve, then -parameter curve of is an asymtotic curve.
Theorem 15. Let be a developable spacelike ruled surface given by the parametrization (2). The s-parameter curve of is also asymtotic curve if and only if is a minimal surface.
Proof. Assume that -parameter curve of the surface an asymtotic curve. Then
where is a unit normal vector field of the surface . Since is a developable ruled surface,
Thus from (34)
Conversely, let be a minimal surface. From , we get since is a developable ruled surface, we obtain This completes the proof.
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