SMARANDACHE CURVES ACCORDING TO CURVES ON A SPACELIKE SURFACE IN MINKOWSKI 3-SPACE R1

In this paper, we introduce Smarandache curves according to the Lorentzian Darboux frame of a curve on spacelike surface in Minkowski 3-space R31. 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.


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][4][5][6][7][8].For instance, the special Smarandache curves according to Darboux frame in E 3 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 3-space R 3  1 .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 3-space and give Lorentzian Darboux frame that will be used throughout the paper.Section 3 is devoted to the study of four Smarandache curves, T-Smarandache curve, T-Smarandache curve, -Smarandache curve, and T-Smarandache curve by considering the relationship with invariants   ,   (), and   () of curve on spacelike surface in Minkowski 3-space R 3 1 .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.

Basic Concepts
The Minkowski 3-space R 3  1 is the Euclidean 3-space R 3 provided with the standard flat metric given by where ( 1 ,  2 ,  3 ) is a rectangular Cartesian coordinate system of R 3 1 .Since ⟨⋅, ⋅⟩ is an indefinite metric, recall that a nonzero vector x ∈ R 3  1 can have one of three Lorentzian causal characters; it can be spacelike if ⟨x, x⟩ > 0, timelike if ⟨x, x⟩ < 0, and null (lightlike) if ⟨x, x⟩ = 0.In particular, the norm (length) of a vector x ∈ R 3  1 is given by ‖x‖ = √|⟨x, x⟩| and two vectors x and y are said to be orthogonal, if ⟨x, y⟩ = 0.
where × is the pseudovector product in the space R 3 1 .
Lemma 2. In the Minkowski 3-space R 3 1 , 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.
1 , () =  and  :  ⊂ R →  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 {T, , } which is called the Lorentzian Darboux frame along the curve  where () = T()×() is a unit spacelike vector.The corresponding Frenet formulae of  read where   () = −⟨T  (), ()⟩,   () = ⟨T  (), ()⟩, and   () = −⟨  (), ()⟩ are the asymptotic curvature, the geodesic curvature, and the principal curvature of  on the surface  in R 3 1 , 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: an asymptotic curve iff   ≡ 0, a geodesic curve iff   ≡ 0, a principal curve iff   ≡ 0. ( Since  is a unit-speed curve, α is perpendicular to T, but α 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 R 3 1 .Let  = () be regular unit speed curves lying fully with the Lorentzian Darboux frame {T, , } on the surface  in R 3  1 .There is not a geodesic curve on .
The pseudosphere with center at the origin and of radius  = 1 in the Minkowski 3-space R 3  1 is a quadric defined by Let  :  ⊂ R →  2 1 be a curve lying fully in pseudosphere  2  1 in R 3 1 .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   () = det((),   (),    ()) 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   () = det((),   (),    ()) 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  2 1 are the null straight lines, which are the null geodesics.

Smarandache Curves according to
Curves on a Spacelike Surface in Minkowski 3-Space R 3 1 In the following section, we define the Smarandache curves according to the Lorentzian Darboux frame in Minkowski 3-space.Also, we obtain the Sabban frame and the geodesic curvature of the Smarandache curves lying on pseudosphere  2 1 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 R 3  1 with the moving Lorentzian Darboux frame {T, , }.Then T-Smarandache curve of  is defined by where ,  ∈ R 0 and  2 −  2 = 2.
Definition 5. Let  = () be a spacelike curve lying fully on the spacelike surface  in R 3 1 with the moving Lorentzian Darboux frame {T, , }.Then T-Smarandache curve of  is defined by where ,  ∈ R 0 and  2 +  2 = 2. Definition 6.Let  = () be a spacelike curve lying fully on the spacelike surface  in R 3 1 with the moving Lorentzian Darboux frame {T, , }.Then -Smarandache curve of  is defined by where ,  ∈ R 0 and  2 −  2 = 2.
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 R 3 1 with the moving Lorentzian Darboux frame {T, , }.Then = 0 for all , then the Sabban frame {, T  ,   } of the T -Smarandache curve  is given by and the geodesic curvature   of the curve  reads where  = sign(  +   ) for all  and (ii) if   +   = 0 for all , then the Sabban frame {, T  ,   } of the T-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   +   ̸ = 0 for all , since   = (/ * )( * /), then the tangent vector T  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 {, T  ,   } of .
(ii) If   +  = 0 for all , then   is null.So, the tangent vector T  of the curve  is a null vector.It is known that the only null curves lying on pseudosphere  2 1 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 {, T  ,   } 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 R 3  1 with the moving Lorentzian Darboux frame {T, , }.Then = 0 for all , then the Sabban frame {, T  ,   } of the T-Smarandache curve  is given by and the geodesic curvature   of the curve  reads where  = sign(2 and the geodesic curvature   of the curve  reads where  = sign((  ) 2 − 2 and the geodesic curvature   of the curve  reads where ) for all  and 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 R 3  1 with the moving Lorentzian Darboux frame {T, , }.Then the T-Smarandache curve  is spacelike and the Sabban frame {, T  ,   } 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) (  ) 2 − 2 2  > 0 for all .Since   = (/ * )( * /), the tangent vector T  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 {, T  ,   } of .
In the theorems which follow, in a similar way as in Theorem 12, we obtain the Sabban frame {, T  ,   } 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 R 3  1 with the moving Lorentzian Darboux frame {T, , }.Then the T-Smarandache curve  is spacelike and the Sabban frame {, T  ,   } is given by and the geodesic curvature   of the curve  reads where and the geodesic curvature   of the curve  reads where  = sign(  −   ) for all  and (ii) if   −   = 0 for all , then the Sabban frame {, T  ,   } 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 (51) Then, there are two following cases.
(i) If   −  ̸ = 0 for all , since   = (/ * )( * /), then we obtain that the unit tangent vector T  of the curve  is a spacelike vector such that where and  = sign(  −   ).
On the other hand, from (17) and (52) it can be easily seen that is a unit timelike vector.
(ii) If   −  = 0 for all , then   is null.So, the tangent vector T  of the curve  is a null vector.It is known that the only null curves lying on pseudosphere  2 1 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 R 3  1 with the moving Lorentzian Darboux frame {T, , }.Then the T-Smarandache curve  is spacelike and the Sabban frame {, T  ,   } is given by and the geodesic curvature   of the curve  reads where where  ∈ (−1, 1).