Abstract
We introduce Smarandache curves according to the Lorentzian Darboux frame of a curve on spacelike surface in Minkowski 3-space . Also, we obtain the Sabban frame and the geodesic curvature of the Smarandache curves and give some characterizations on the curves when the curve α is an asymptotic curve or a principal curve. And we give an example to illustrate these curves.
1. Introduction
In the theory of curves in the Euclidean and Minkowski spaces, one of the interesting problems is the characterization of a regular curve. In the solution of the problem, the curvature functions and of a regular curve have an effective role. It is known that the shape and size of a regular curve can be determined by using its curvatures and . Another approach to the solution of the problem is to consider the relationship between the corresponding Frenet vectors of two curves. For instance, Bertrand curves and Mannheim curves arise from this relationship. Another example is the Smarandache curves. They are the objects of Smarandache geometry, that is, a geometry which has at least one Smarandachely denied axiom [1]. The axiom is said to be Smarandachely denied if it behaves in at least two different ways within the same space. Smarandache geometries are connected with the theory of relativity and the parallel universes.
By definition, if the position vector of a curve is composed by the Frenet frame’s vectors of another curve , then the curve is called a Smarandache curve [2]. Special Smarandache curves in the Euclidean and Minkowski spaces are studied by some authors [3–8]. For instance, the special Smarandache curves according to Darboux frame in are characterized in [9].
In this paper, we define Smarandache curves according to the Lorentzian Darboux frame of a curve on spacelike surface in Minkowski -space . Inspired by the previous papers we investigate the geodesic curvature and the Sabban frame’s vectors of Smarandache curves. In Section 2, we explain the basic concepts of Minkowski -space and give Lorentzian Darboux frame that will be used throughout the paper. Section 3 is devoted to the study of four Smarandache curves, -Smarandache curve, -Smarandache curve, -Smarandache curve, and -Smarandache curve by considering the relationship with invariants , , and of curve on spacelike surface in Minkowski -space . Also, we give some characterizations on the curves when the curve is an asymptotic curve or a principal curve. Finally, we illustrate these curves with an example.
2. Basic Concepts
The Minkowski 3-space is the Euclidean 3-space provided with the standard flat metric given by where is a rectangular Cartesian coordinate system of . Since is an indefinite metric, recall that a nonzero vector can have one of three Lorentzian causal characters; it can be spacelike if , timelike if , and null (lightlike) if . In particular, the norm (length) of a vector is given by and two vectors and are said to be orthogonal, if . For any and in the space , the pseudovector product of and is defined by
Next, recall 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 every [10]. If for every , then is a regular curve in . A spacelike (timelike) regular curve is parameterized by pseudoarclength parameter which is given by ; then the tangent vector along has unit length; that is, for all , respectively.
Remark 1. Let , , and be vectors in . Then where is the pseudovector product in the space .
Lemma 2. In the Minkowski 3-space , the following properties are satisfied [10]:(i)two timelike vectors are never orthogonal;(ii)two null vectors are orthogonal if and only if they are linearly dependent;(iii)timelike vector is never orthogonal to a null vector.
Let , and be a spacelike embedding and a regular curve, respectively. Then we have a curve on the surface which is defined by and since is a spacelike embedding, we have a unit timelike normal vector field along the surface which is defined by Since is a spacelike surface, we can choose a future directed unit timelike normal vector field along the surface . Hence we have a pseudoorthonormal frame which is called the Lorentzian Darboux frame along the curve where is a unit spacelike vector. The corresponding Frenet formulae of read where , , and are the asymptotic curvature, the geodesic curvature, and the principal curvature of on the surface in , respectively, and is arclength parameter of . In particular, the following relations hold: Both and may be positive or negative. Specifically, is positive if curves towards the normal vector , and is positive if curves towards the tangent normal vector . Also, the curve is characterized by , , and as follows: Since is a unit-speed curve, is perpendicular to , but may have components in the normal and tangent normal directions: These are related to the total curvature of by the formula From (9) we can give the following proposition.
Proposition 3. Let be a spacelike surface in . Let be regular unit speed curves lying fully with the Lorentzian Darboux frame on the surface in . There is not a geodesic curve on .
The pseudosphere with center at the origin and of radius in the Minkowski 3-space is a quadric defined by Let be a curve lying fully in pseudosphere in . Then its position vector is a spacelike, which means that the tangent vector can be spacelike, timelike, or null. Depending on the causal character of , we distinguish the following three cases [5].
Case 1 ( is a unit spacelike vector). Then we have orthonormal Sabban frame along the curve , where is the unit timelike vector. The corresponding Frenet formulae of according to the Sabban frame read where is the geodesic curvature of and is the arclength parameter of . In particular, the following relations hold:
Case 2 ( is a unit timelike vector). Hence we have orthonormal Sabban frame along the curve , where is the unit spacelike vector. The corresponding Frenet formulae of according to the Sabban frame read where is the geodesic curvature of and is the arclength parameter of . In particular, the following relations hold:
Case 3 ( is a null vector). It is known that the only null curves lying on pseudosphere are the null straight lines, which are the null geodesics.
3. Smarandache Curves according to Curves on a Spacelike Surface in Minkowski 3-Space
In the following section, we define the Smarandache curves according to the Lorentzian Darboux frame in Minkowski -space. Also, we obtain the Sabban frame and the geodesic curvature of the Smarandache curves lying on pseudosphere and give some characterizations on the curves when the curve is an asymptotic curve or a principal curve.
Definition 4. Let be a spacelike curve lying fully on the spacelike surface in with the moving Lorentzian Darboux frame. Then -Smarandache curve of is defined by where and .
Definition 5. Let be a spacelike curve lying fully on the spacelike surface in with the moving Lorentzian Darboux frame. Then -Smarandache curve of is defined by where and .
Definition 6. Let be a spacelike curve lying fully on the spacelike surface in with the moving Lorentzian Darboux frame. Then -Smarandache curve of is defined by where and .
Definition 7. Let be a spacelike curve lying fully on the spacelike surface in with the moving Lorentzian Darboux frame. Then -Smarandache curve of is defined by where and .
Thus, there are two following cases.
Case 4 ( is an asymptotic curve). Then, we have the following theorems.
Theorem 8. Let be an asymptotic spacelike curve lying fully on the spacelike surface in with the moving Lorentzian Darboux frame. Then(i)if for all , then the Sabban frame of the -Smarandache curve is given by and the geodesic curvature of the curve reads where for all and (ii)if for all , then the Sabban frame of the -Smarandache curve is a null geodesic.
Proof. We assume that the curve is an asymptotic curve. Differentiating (15) with respect to and using (5) we obtain Then, there are two following cases.(i)If for all , since , then the tangent vector of the curve is a spacelike vector such that where On the other hand, from (15) and (23) it can be easily seen that is a unit timelike vector. Consequently, the geodesic curvature of the curve is given by From (15), (23), and (25) we obtain the Sabban frame of .(ii)If for all , then is null. So, the tangent vector of the curve is a null vector. It is known that the only null curves lying on pseudosphere are the null straight lines, which are the null geodesics.
In the theorems which follow, in a similar way as in Theorem 8, we obtain the Sabban frame and the geodesic curvature of a spacelike Smarandache curve. We omit the proofs of Theorems 9, 10, and 11, since they are analogous to the proof of Theorem 8.
Theorem 9. Let be an asymptotic spacelike curve lying fully on the spacelike surface in with the moving Lorentzian Darboux frame. Then(i)if for all , then the Sabban frame of the -Smarandache curve is given by and the geodesic curvature of the curve reads where for all and (ii)if for all , then the Sabban frame of the -Smarandache curve is a null geodesic.
Theorem 10. Let be an asymptotic spacelike curve lying fully on the spacelike surface in with the moving Lorentzian Darboux frame. Then(i)if for all , then the Sabban frame of the -Smarandache curve is given by and the geodesic curvature of the curve reads where for all and (ii)if for all , then the Sabban frame of the -Smarandache curve is a null geodesic.
Theorem 11. Let be an asymptotic spacelike curve lying fully on the spacelike surface in with the moving Lorentzian Darboux frame. Then(i)if for all , then the Sabban frame of the -Smarandache curve is given by and the geodesic curvature of the curve reads where for all and (ii)if for all , then the Sabban frame of the -Smarandache curve is a null geodesic.
Case 5 ( is a principal curve). Then, we have the following theorems.
Theorem 12. Let be a principal spacelike curve lying fully on the spacelike surface in with the moving Lorentzian Darboux frame. Then the -Smarandache curve is spacelike and the Sabban frame is given by and the geodesic curvature of the curve reads where
Proof. We assume that the curve is a principal curve. Differentiating (15) with respect to and using (5) we obtain
where from (9) for all . Since , the tangent vector of the curve is a spacelike vector such that
where
On the other hand, from (15) and (40) it can be easily seen that
is a unit timelike vector.
Consequently, the geodesic curvature of the curve is given by
where
From (15), (40), and (42) we obtain the Sabban frame of .
In the theorems which follow, in a similar way as in Theorem 12, we obtain the Sabban frame and the geodesic curvature of a spacelike Smarandache curve. We omit the proofs of Theorems 13 and 15, since they are analogous to the proof of Theorem 12.
Theorem 13. Let be a principal spacelike curve lying fully on the spacelike surface in with the moving Lorentzian Darboux frame. Then the -Smarandache curve is spacelike and the Sabban frame is given by and the geodesic curvature of the curve reads where
Theorem 14. Let be a principal spacelike curve lying fully on the spacelike surface in with the moving Lorentzian Darboux frame . Then (i)if for all , the -Smarandache curve is spacelike and the Sabban frame is given by and the geodesic curvature of the curve reads where for all and (ii)if for all , then the Sabban frame of the -Smarandache curve is a null geodesic.
Proof. We assume that the curve is a principal curve. Differentiating (17) with respect to and using (5) we obtain Then, there are two following cases.(i)If for all , since , then we obtain that the unit tangent vector of the curve is a spacelike vector such that where and . On the other hand, from (17) and (52) it can be easily seen that is a unit timelike vector. Consequently, the geodesic curvature of the curve is given by From (17), (52), and (54) we obtain the Sabban frame of .(ii)If for all , then is null. So, the tangent vector of the curve is a null vector. It is known that the only null curves lying on pseudosphere are the null straight lines, which are the null geodesics.
Theorem 15. Let be a principal spacelike curve lying fully on the spacelike surface in with the moving Lorentzian Darboux frame. Then the -Smarandache curve is spacelike and the Sabban frame is given by and the geodesic curvature of the curve reads where
Example 16. Let us define a spacelike ruled surface (see Figure 1) in the Minkowski -space such as
where .
Then we get the Lorentzian Darboux frame along the curve as follows:
where is spacelike vectors and is a unit timelike vector.
Moreover, the geodesic curvature , the asymptotic curvature , and the principal curvature of the curve have the form
Taking , and using (15), we obtain that the -Smarandache curve of the curve is given by (see Figure 2(a))
Taking and using (16), we obtain that the -Smarandache curve of the curve is given by (see Figure 2(b))
Taking , and using (17), we obtain that the -Smarandache curve of the curve is given by (see Figure 3(a))
Taking , , and and using (18), we obtain that the -Smarandache curve of the curve is given by (see Figure 3(b))
Also, the Smarandache curves on of for with Figure 4 are shown.
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(b)
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Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.