Abstract
We introduce special Smarandache curves based on Sabban frame on and we investigate geodesic curvatures of Smarandache curves on de Sitter and hyperbolic spaces. The existence of duality between Smarandache curves on de Sitter space and Smarandache curves on hyperbolic space is shown. Furthermore, we give examples of our main results.
1. Introduction
Curves as a subject of differential geometry have been intriguing for researchers throughout mathematical history and so they have been one of the interesting research fields. Regular curves play a central role in the theory of curves in differential geometry. In the theory of curves, there are some special curves such as Bertrand curves, Mannheim curves, involute and evolute curves, and pedal curves in which differential geometers are interested. A common approach to characterization of curves is to consider the relationship between the corresponding Frenet vectors of two curves. Bertrand and Mannheim curves are excellent examples for such cases. In the study of fundamental theory and the characterizations of space curves, the corresponding relations between the curves are a very fascinating problem. Recently, a new special curve is named according to the Sabban frame in the Euclidean unit sphere; Smarandache curve has been defined by Turgut and Yilmaz in Minkowski space-time [1]. Ali studied Smarandache curves with respect to the Sabban frame in Euclidean 3-space [2]. Then Taşköprü and Tosun studied Smarandache curves on [3]. Smarandache curves have also been studied by many researchers [1, 4–7]. Smarandache curves are one of the most important tools in Smarandache geometry. Smarandache geometry has an important role in the theory of relativity and parallel universes. There are many results related to Smarandache curves in Euclidean and Minkowski spaces, but Smarandache curves are getting more tedious and complicated when de Sitter space is concerned. A regular curve in Minkowski space-time, whose position vector is associated with Frenet frame vectors on another regular curve, is called a Smarandache curve [1].
In this paper, we define Smarandache curves on de Sitter surface according to the Sabban frame in Minkowski 3-space. We obtain the geodesic curvatures and the expressions for the Sabban frame’s vectors of special Smarandache curves on de Sitter surface. Furthermore, we give some examples of special de Sitter and hyperbolic Smarandache curves in Minkowski 3-space.
2. Preliminaries
In this section, we prepare some definitions and basic facts. For basic concepts and details of properties, see [8, 9]. Consider as a three-dimensional vector space. For any vectors and in the pseudoscalar product of and is defined by . It is called Minkowski 3-space. Recall that a nonzero vector is spacelike if , timelike if , and null (lightlike) if . The norm (length) of a vector is given by and two vectors and are said to be orthogonal if . Next, we say that an arbitrary curve in can locally be spacelike, timelike, or null (lightlike) if all of its velocity vectors are, respectively, spacelike, timelike, or null (lightlike) for all . If for every , then is a regular curve in . A spacelike (timelike) regular curve is parameterized by a pseudoarclength parameter which is given by , and then the tangent vector along has unit length; that is, for all .
Let , . The Lorentzian vector cross-product is defined as follows: and also the following relations hold:(i),(ii),where , , .
We now define de Sitter 2-space by and hyperbolic space in Minkowski 3-space by We can express a new frame different from the Frenet frame for a regular curve. Let be a regular unit speed curve lying fully on . Then its position vector is spacelike, which implies that the tangent vector is the unit timelike, spacelike, or null vector for all .
In our work, we are concerned with the vector which may be the unit timelike or spacelike.
Let be a regular unit speed curve lying fully on for all and its position vector a unit spacelike vector; then is a unit timelike and so is a unit spacelike vector. In this case, the curve is called a timelike curve. If is a unit spacelike vector, then is a unit timelike vector. In this case, the curve is called a spacelike curve and we have an orthonormal Sabban frame along the curve , where is the unit spacelike or timelike vector. Then Frenet formulas of are given by where , the curve is timelike for and spacelike for , and is the geodesic curvature of on , which is given by , where is the arc length parameter of . This relation is also given by [10, 11] for . In particular, by using equation (ii), the following relations hold:
Definition 1. A unit speed regular curve lying fully in Minkowski 3-space, whose position vector is associated with Sabban frame vectors on another regular curve , is called a Smarandache curve [1].
Based on this definition, if a regular unit speed curve is lying fully on for all and its position vector is a unit spacelike, then the Smarandache curve of curve is a regular unit speed curve lying fully on or . In this case we have the following:(a)the Smarandache curve may be a timelike curve on ,(b)the Smarandache curve may be a spacelike curve on , or(c)the Smarandache curve is in for all .Let and be the moving Sabban frames of and , respectively. Then we have the following definitions and theorems of Smarandache curves given in Section 3. In Section 3, we deal with Smarandache curves on de Sitter and hyperbolic spaces for timelike curves. Similar results are given for spacelike curves in the Appendix.
3. De Sitter and Hyperbolic Smarandache Curves for Timelike Curves
In this section we give different Smarandache curves on de Sitter and hyperbolic spaces in Minkowski-space. Let be a timelike curve on ; then the Smarandache partner curve of is either timelike/spacelike or hyperbolic curve. We refer to the hyperbolic Smarandache curve of a timelike curve as the hyperbolic duality of .
To avoid repetition we use in the following theorems in this section. If we take , then the Smarandache curve is timelike or spacelike, and if we take , then is hyperbolic.
Definition 2. Let be a unit speed regular timelike curve lying fully on . The curve of defined by is called the -Smarandache curve of and fully lies on , where and . If ; then the hyperbolic -Smarandache curve is undefined since the equation has no solution in .
Theorem 3. Let be a regular unit speed timelike curve lying fully on with the Sabban frame and geodesic curvature . If is the -timelike Smarandache curve of , then the relationships between the Sabban frames of and its -Smarandache curve are given by where and its geodesic curvature is given by
Proof. By taking the derivative of (6) with respect to and by using (4), we get or equivalently where Hence, the unit timelike tangent vector of the curve is given by where if for all and if for all . From (6) and (12) we get It is easily seen that is a unit spacelike vector. On the other hand, differentiating (12) with respect to , we find and by combining (11) and (14) we have Consequently, the geodesic curvature of the curve is given by
Corollary 4. Let be a regular unit speed timelike curve lying fully on . Then the -spacelike Smarandache curve of does not exist.
Definition 5. Let be a regular unit speed timelike curve lying fully on . Then the -Smarandache curve of is defined by where and .
Theorem 6. Let be a regular unit speed timelike curve lying fully on with the Sabban frame and geodesic curvature . If is the -timelike (hyperbolic) Smarandache curve of , then its frame is given byThe geodesic curvature of curve is given by where
Proof. We take . By taking the derivative of (17) with respect to and by using (4), we get or equivalently Taking the Lorentzian inner product in (22) we have If , then is a timelike vector. So Therefore, the unit timelike tangent vector of the curve is given by On the other hand, from (17) and (25) it can be easily seen that is a unit spacelike vector. Differentiating (25) with respect to , we obtain where and by combining (24) and (26) we get Consequently, we have The proof of case is similar.
The following corollary is proved by the same methods as the above theorem.
Corollary 7. Let be a regular unit speed timelike curve lying fully on with the Sabban frame and geodesic curvature . If is the -spacelike Smarandache curve of , then its frame is given by The geodesic curvature of curve is given by where , and can be calculated as in Theorem 6.
Definition 8. Let be a regular unit speed timelike curve lying fully on . Then the -Smarandache curve of is defined by where and .
Theorem 9. Let be a regular unit speed timelike curve lying fully on with the Sabban frame and geodesic curvature . If is the -timelike (hyperbolic) Smarandache curve of , then its frame is given by The geodesic curvature of curve is given by where
Proof. Let . By taking the derivative of (33) with respect to and by using (4), we get or equivalently Taking the Lorentzian inner product in (38) we have and is a unit timelike vector for . It follows that Therefore, the unit timelike tangent vector of the curve is given by On the other hand, from (33) and (41) it can be easily seen that is a unit spacelike vector. Differentiating (41) with respect to , we find where and by combining (40) and (43) we get As a result, we have The proof of case is similar.
Corollary 10. Let be a regular unit speed timelike curve lying fully on with the Sabban frame and geodesic curvature . If is the -spacelike Smarandache curve of , then its frame is given by The geodesic curvature of curve is given by where , and can be calculated as in Theorem 9.
Definition 11. Let be a regular unit speed timelike curve lying fully on . Then the -Smarandache curve of is defined by where and .
Theorem 12. Let be a regular unit speed timelike curve lying fully on with the Sabban frame and geodesic curvature . If is the -timelike (hyperbolic) Smarandache curve of , then its frame , is given bywhere . If we take , then the Smarandache curve is timelike or hyperbolic, respectively. Furthermore, the geodesic curvature of curve is given by where
Proof. We take . By taking the derivative of (49) with respect to and using (4), we get
or equivalently
Taking the Lorentzian inner product in (54) we have
For , is a unit timelike vector. It follows that
Therefore, the unit timelike tangent vector of the curve is given by
On the other hand, taking the cross-product of (49) with (57) it can be easily seen that
This means that the is a unit spacelike vector. In order to obtain the tangent vector of let us differentiate (57) with respect to . We find
where
and by combining (56) and (59) we get
Finally, the geodesic curvature of the curve is given by
The proof of case is similar.
Corollary 13. Let be a regular unit speed timelike curve lying fully on with the Sabban frame and geodesic curvature . If is the -spacelike Smarandache curve of , then, for , its frame is given byFurthermore, the geodesic curvature of curve is given by where , and can be calculated as in Theorem 12.
Example 14. Let us consider a unit speed timelike curve on defined by
Then the orthonormal Sabban frame of can be calculated as follows:
The geodesic curvature of is . In terms of the definitions, we obtain Smarandache curves according to Sabban frame on .
Firstly, when we take and , then the timelike -Smarandache curve is given by
and the Sabban frame of the -Smarandache curve is given by
and its geodesic curvature is . Here the hyperbolic -Smarandache curve is undefined.
Secondly, when we take and , then the timelike -Smarandache curve is given by
and if we take and , then the hyperbolic -Smarandache curve is given by
Thirdly, when we take and , then the spacelike -Smarandache curve is given by
and if we take and , then the hyperbolic -Smarandache curve is given by
Finally, when we take , , and , then the -Smarandache curve is a timelike curve and given by
and if we take , , and , then the -Smarandache curve is a hyperbolic curve and given by
On the other hand, in the last case, if we take , , and (i.e., ), then the -Smarandache curve is spacelike and given by
The Sabban frames and geodesic curvatures of , , and -Smarandache curves can be easily obtained by using a similar way to the above. Also we give the curve and its Smarandache partners in Figure 1.
Appendix
In this section, we give as a table different Smarandache curves on de Sitter space or on hyperbolic space for spacelike curves in Minkowski-space. We give the theorem about undefined Smarandache curve below. The other , and -Smarandache curves on and -Smarandache curves on and their corresponding Sabban frames and geodesic curvatures are similar to those in the previous section.
Theorem A.1. Let be a regular unit speed spacelike curve lying fully on . Then the -Smarandache curve of does not exist.
Proof. Let be a regular unit speed spacelike curve lying fully on . Then the -Smarandache curve of can be written as follows: , and which is contradiction.
The Smarandache curves of a regular unit spacelike curve are given in Table 1.
Example A.2. Let us consider a unit speed spacelike curve on defined by
Then the orthonormal Sabban frame of can be calculated as follows:
The geodesic curvature of is expressed as
In terms of the definitions, we obtain Smarandache curves according to Sabban frame on . Firstly, we take and ; then the spacelike -Smarandache curve is given by
and also when we take and , then the hyperbolic -Smarandache curve is given by
The Sabban frames and geodesic curvatures are similar to the above section. Secondly, we take and ; then the spacelike Smarandache- curve is given by
Here the hyperbolic -Smarandache curve is undefined.
Thirdly, we take and ; then the spacelike -Smarandache curve is given by
and also when we take and , then the hyperbolic -Smarandache curve is given by
Finally, when , , and , then the -Smarandache curve is a spacelike curve and is given by
and also when we take , , and , then the hyperbolic -Smarandache curve is given by
The Sabban frames and geodesic curvatures of the , , and -Smarandache curves can be easily obtained by using methods similar to those in the previous section. Furthermore, we give curve the and its Smarandache partners in Figure 2.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.