Eikonal slant helices and eikonal Darboux helices in 3-dimensional pseudo-riemannian manifolds

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1 Eikonal slant helices and eikonal Darboux helices in -dimensional pseudo-riemannian maniolds Mehmet Önder a, Evren Zıplar b a Celal Bayar University, Faculty o Arts and Sciences, Department o Mathematics, Muradiye Campus, Muradiye, Manisa, Turkey. mehmet.onder@cbu.edu.tr b Çankırı Karatekin University, Faculty o Science, Department o Mathematics, Çankırı, Turkey evrenziplar@karatekin.edu.tr Abstract In this study, we give deinitions and characterizations o eikonal slant helices, eikonal Darboux helices and non-normed eikonal Darboux helices in -dimensional pseudo- Riemannian maniold M. We show that every eikonal slant helix is also an eikonal Darboux helix or timelike and spacelike curves. Furthermore, we obtain that i the non-null curve α is a non-normed eikonal Darboux helix, then α is an eikonal slant helix i and only i εκ + ετ = constant, where κ and τ are curvature and torsion o α, respectively. Finally, we deine null-eikonal helices, slant helices and Darboux helices. Also, we give their characterizations. MSC: 5B40, 5C50. Key words: Eikonal slant helix; eikonal Darboux helix; null slant helix.. Introduction In the nature and science, some special curves have an important role and many applications. The well-known o such curves is helix curve. In the Euclidean -space E, a general helix is deined as a special curve whose tangent line makes a constant angle with a ixed straight line which is called the axis o the helix [4]. This deinition gives that the tangent indicatrix o a general helix is a planar curve. Moreover, the classical result or the helices irst was given by Lancret in 80 and proved by B. de Saint enant in 845 as ollows: A necessary and suicient condition that a curve to be a general helix is that the ratio o the irst curvature to the second curvature be constant i.e., κ / τ is constant along the curve, where κ and τ denote the irst and second curvatures o the curve, respectively [0]. The same deinition is also valid in Lorentzian space and spacelike, timelike and null helices have been studied by some mathematicians [7-9]. Furthermore, there exist more special curves in the space such as slant helix which irst introduced by Izumiya and Takeuchi by the property that the normal lines o curve make a constant angle with a ixed direction in the Euclidean -space E [4]. Slant helices have been studied by some mathematicians and new kinds o these curves also have been introduced [,,6,7,9]. Moreover, these curves have been considered in Lorentzian spaces [,].

2 Later, a new kind o helices has been deined by Zıplar, Şenol and Yaylı according to the Darboux vector o a space curve in E. They have called this new curve as Darboux helix which is deined by the property that the Darboux vector o a space curve makes a constant angle with a ixed direction and they have given the characterizations o this new special curve []. Let M be a Riemannian maniold with the metric g and : M R be a unction with gradient applications o. The unction is called eikonal i is constant [5]. There exist many in mathematical physics and geometry. For instance, i is non-constant on connected M, then the Riemannian condition = is precisely the eikonal equation o geometrical optics. So, on a connected M, a non-constant real valued unction is Riemannian i satisies this eikonal equation. In the geometrical optical interpretation, the level sets o are interpreted as wave ronts. The characteristics o the eikonal equation (as a partial dierential equation), are then the solutions o the gradient low equation or (an ordinary dierential equation), x =, which are geodesics o M orthogonal to the level sets o, and which are parameterized by arc length. These geodesics can be interpreted as light rays orthogonal to the wave ronts (See [0] or details). Later, Şenol, Zıplar and Yaylı have deined eikonal helices and eikonal slant helices by considering a space curve with a unction : M R where M is a Riemannian maniold []. In this study, we deine and give the characterizations o -eikonal slant helices and -eikonal Darboux helices or non-null and null curves in a pseudo-riemannian maniold. For this purpose, we need the ollowing deinitions. Deinition.. ([8]) A metric tensor g in a smooth maniold M is a symmetric nondegenerate (0, ) tensor ield in M. On the other hand i TM is the tangent bundle o M, then or all X, Y TM, g( X, Y ) = g( Y, X ) and at each point p o M, i g( X, Y ) = 0 or all Y T ( M ), then X = 0 (non-degenerate) where T ( M ) is the tangent space o M at the point p and p g : T ( M ) T ( M ) R. p p p p p p p Deinition.. ([8]) A pseudo-riemannian maniold (or semi-riemannian maniold) is a smooth maniold M urnished with a metric tensor g. That is, a pseudo-riemannian maniold is an ordered pair ( M, g ). Deinition.. Let M be a pseudo-riemannian maniold and g be its metric. For the unction : M R, it is said that is eikonal i is constant, where is gradient o d ( X ) = g, X = X ( )., i.e., ( )

3 Lemma.. ([8]) Let ( M, g ) be a pseudo-riemannian maniold and be the Levi-Civita connection o M. The Hessian that H ( X, Y ) = g( (grad ), Y ), X H o a F( M ) where F( M ) shows the set o dierentiable unctions deined on M. From Lemma., we have the ollowing corollary. is the symmetric (0,) tensor ield such Corollary.. The Hessian parallel in M. H o a F( M ) is zero, i.e., H = 0 i and only i is. Non-null Eikonal Slant Helices and Non-null Eikonal Darboux Helices Let ( M, g ) be a time-oriented -dimensional pseudo-riemannian maniold and α : Ι M be a unit speed curve on M, i.e., g( α, α ) = ε = ± is satisied along α where α is the velocity vector iled o the curves and g shows the metric tensor (or Lorentzian metric) given by g( a, b) = ab + ab + ab or the vectors a = ( a, a, a), b = ( b, b, b ) TM. The constant ε = ± deined by ε = g( α, α ) is called the causal character o α. Then, a unit speed curve α is said to be spacelike or timelike i its causal character is or -, respectively. The curve α is said to be a Frenet curve i g( α, α ) 0. Like Euclidean geometry, every Frenet curve α on ( M, g ) admits an orthonormal Frenet rame ield {,, } along α such that = α ( s). The vector ields,, are called tangent vector ield, principle normal vector ield and binormal vector ield o α, respectively,, satisies the ollowing Frenet-Serret ormula: and { } 0 ε κ 0 = ε κ 0 ε τ, 0 0 ε τ where is the Levi-Civita connection o ( M, g ) [,,5]. The unctions κ 0 and τ are called the curvature and torsion, respectively. The constants ε and ε are deined by ε i = g( i, i ), i =,. and called second causal character and third causal character o α, respectively. Note that ε = ε ε and i j = εiε jk, where ( i, j, k ) = (,,), (,,), (,,). The vector W = τ κ is called Darboux vector o the curve α. Then or the Frenet ormulae we have = W, ( i =,,) ; where " " shows the vector product in M. i i As in the case o Riemannian geometry, a Frenet curve α is a geodesic i and only i κ = 0. A circular helix is a Frenet curve whose curvature and torsion are constants. I the curvature κ is constant and the torsion τ is zero, then the curve is called a pseudo circle.

4 Pseudo circles are regarded as degenerate helices. Helices, which are not circles, are requently called proper helices. Deinition.. Let M be a -dimensional pseudo-riemannian maniold with the Lorentzian metric g and let α ( s) be a non-null Frenet curve with the Frenet rame {,, } in M. Let : M R be an eikonal unction along curve α, i.e. = constant along the curve α. I the unction g (, ) is a non-zero constant along α, then α is called a non-null -eikonal slant helix. And, is called the axis o the - eikonal slant helix α. Deinition.. Let α be a non-null Frenet curve in M be a pseudo-riemannian maniold with the Lorentzian metric g and and Darboux vector W = τ κ. Also, let the unit Darboux vector W τ κ =, 0 εκ + ετ εκ + ετ M with Frenet rame {,, }, non-zero curvatures κ,τ : M R be an eikonal unction along α. I o the curve α makes a constant angle ϕ with the gradient o the unction, that is ( ) g W helix. 0, is constant along α, then the curve α is called a non-null -eikonal Darboux Especially, i g ( W ), = constant, then α is called a non-normed non-null -eikonal Darboux helix. Then, we have the ollowing Corollary. Corollary.. A non-normed non-null -eikonal Darboux helix is a non-null -eikonal Darboux helix i and only i ε κ + ε τ is constant. Example.. We consider the pseudo-riemannnian maniold metric g. Let M =R with the Lorentzian : M R (,, ) (,, ) x y z x y z = x + y + z be a unction deined in M and consider the spacelike curve

5 in α : I R M s s bs s α ( s) = a cosh, a sinh, ; a, b > 0 a + b a + b a + b M. I we compute, we ind out ( ) = + 4 x + y, and, along the curve α, we ind out as = ( x, y,). Then, we have = 4a = constant. That is, is an eikonal unction along α. Moreover, by a simple computation we have that the principal normal o the curve is Since s s ( s) = cosh, sinh,0. a + b a + b s s = a cosh, asinh,, a + b a + b along α, we easily see that ( ) -eikonal slant helix in M. g, = a = constant which means that α is a non-null On the other hand, non-normed Darboux vector o α is W a( a + b ) + ab s a( a + b ) + ab s = sinh, cosh, / / b( a b ) b( a b ) + a + b + a + b a ( a + b ) / ( a + b ) and curvatures are a, a + κ = τ = b a + b b a ( a + b ) g (, W ) = = constant, / ( a + b ), respectively. Then we obtain that along α. So, α is a non-null -eikonal non-normed Darboux helix curve in κ, τ are constants α is also a non-null -eikonal Darboux helix curve in M. M. Since

6 Now, we give some theorems concerned with non-null -eikonal slant helices and - eikonal Darboux helices in pseudo-riemannian maniold. Whenever we write M, we will consider M as a -dimensional pseudo-riemannian maniold with the Lorentzian metric g. Theorem.. Let and assume that α ( s) is not a helix. Let R be a non-null curve in M with non-zero curvatures κ, τ : M R be an eikonal unction along curve α and the Hessian H = 0. I α( s) is a non-null -eikonal slant helix curve in ollowing properties hold: i) The unction κ ( ετ + εκ ) is a real constant. M, then the τ, () κ ii) The axis o -eikonal slant helix is obtained as nτ nκ = + c, ε τ ε κ ε τ ε κ + + where c and n are non-zero constants. Proo. i) Since α is a non-null -eikonal slant helix, we have ( ) g, = c = constant. So, there exist smooth unctions a = a ( s), a = a ( s) = c and a ( ) = a s o arc length s such that = a + c + a, () is a basis o where {,, } From Corollary., TM (tangent bundle o is parallel in M, i.e., derivative in each part o () in the direction in M ). = 0 along α. Then, i we take the M and use the Frenet equations, we get ( [ a ] ε κc) + ( ε a κ + ε a τ ) + ( [ a ] ε τc) = 0, () Since [ a ] = a s i = in () and the Frenet rame {,, } i i ( ), (,,) independent, we have is linearly a εκ c = 0, aκ + aτ = 0, a ετ c = 0. (4) From the second equation o the system (4) we obtain

7 τ a = a. (5) κ Since is an eikonal unction along α, we have is constant. Then () and (5) give that ε ε ε κ τ + a + c = constant and rom (6) we can write ε κ τ + ε a = n, (6), (7) where n is a constant. Since α is not a helix curve in M and curvatures are not zero, we have that n is a non-zero constant. Then, rom (7) we have a = ± n τ ε + ε κ. (8) By taking the derivative o (8) with respect to s and using the third equation o the system (4), we get that the unction κ ( ετ + εκ ) is a constant, which is desired unction. τ, (9) κ ii) By direct calculation rom (5) and (8), we have a = nτ ε τ + ε κ a, = nκ ε τ + ε κ, where n is a non-zero constant. Then, rom () the axis o -eikonal slant helix is nτ nκ = + c. (0) ε τ ε κ ε τ ε κ + + The above Theorem has the ollowing corollary. Corollary.. Let and assume that α ( s) is not a helix. Let R be a non-null curve in M with non-zero curvatures κ, τ : M R be an eikonal unction along curve α

8 and the Hessian H = 0. I α ( s) is a non-null -eikonal slant helix curve in curvatures κ and τ satisy the ollowing non-linear equation system: M, then, the nτ εκ c = 0, ετ + εκ nκ ετ c = 0. ετ + εκ () Theorem.. Let R be a non-null curve in and assume that α ( s) is not a helix. Let M with non-zero curvatures κ, τ : M R be an eikonal unction along curve α and the Hessian H = 0. Then, every non-null -eikonal slant helix in -eikonal Darboux helix in M. Proo. Let α be a non-null -eikonal slant helix in o α is + + M is also a non-null M. Then, rom Theorem., the axis nτ nκ = + c. () ε τ ε κ ε τ ε κ Considering the unit Darboux vector W 0, equality () can be written as ollows = nw0 + c, () which shows that lies on the plane spanned by W 0 and. Since n is a non-zero g, W = n is constant along α, i.e, α is a non-null - constant, rom (), we have ( ) eikonal Darboux helix in M. 0 Theorem.. Let R be a non-null curve in and assume that α ( s) is not a helix. Let M with non-zero curvatures κ, τ : M R be an eikonal unction along curve α and the Hessian H = 0. Let α be a non-normed non-null -eikonal Darboux helix with Darboux vector W. Then α is a non-null -eikonal slant helix i and only i W is a nonzero constant. Proo. Since α is a non-normed non-null ( ) -eikonal Darboux helix, we have g W, = constant. On the other hand, there exist smooth unctions a = a ( s), a = a ( s) and a ( ) = a s o arc length s such that = a + a + a, (4)

9 where,, a a a are assumed non-zero and {,, }.., is parallel in M, i.e., part o (4) in the direction in is a basis o TM. From Corollary = 0 along α. Then, i we take the derivative in each M and use the Frenet equations, we get ( a ε a κ ) + ( ε a κ + a + ε a τ ) + ( a ε a τ ) = 0, (5) where a ( s) = [ a ], ( i =,,). Since the Frenet rame {,, } we have i i is linearly independent, a εκ a = 0, a + ε κa + ε τ a = 0, a ετ a = 0. Equality g ( W ), = constant gives that (6) εaτ εaκ = constant. (7) Dierentiating (7) and using the irst and third equations o system (6) we obtain ε τ ε κ = (8) a a 0 From (8) and the second equation o system (6) it ollows a ε ε ε κ + τ τ ( ) = a (9) In (9) i a = 0, rom (6) we have a = a = 0, i.e., = 0 which is a contradiction. Then we have a 0 and rom (9) we see that a ( ) = a s is constant i and only i ε ε κ + = which means that τ constant εκ constant eikonal slant helix i and only i W is a non-zero constant. ε τ + =, i.e, α is a non-null - From Theorem. and Corollary., we have the ollowing corollary. Corollary.. Let R be a non-null curve in and assume that α ( s) is not a helix. Let M with non-zero curvatures κ, τ : M R be an eikonal unction along curve α and the Hessian H = 0. Let α be a non-normed non-null -eikonal Darboux helix. Then α is a non-null - eikonal slant helix i and only i α is a non-null -eikonal Darboux helix.. Null Eikonal Helices, Slant Helices and Darboux Helices Let α be a curve in -dimensional pseudo-riemannian maniold ( M, g ). Then, the curve α is called a null curve i (, ) 0 g =. By a Cartan rame or Frenet rame {,, } o α,

10 we mean a amily o vector ields = ( s), = ( s), = ( s) along the curve α satisying the ollowing conditions: α ( s) =, g(, ) = g(, ) = 0, g(, ) =, g(, ) = g(, ) = 0, g(, ) =, (0) =, =, =. ([6]). Here, and are called tangent vector ield, binormal vector ield and (principal) normal vector ield o α, respectively. Then the derivative ormula o the rame is given as ollows 0 0 κ 0 0 τ =, () 0 τ κ where κ and τ are called the curvature and torsion o γ, respectively [6]. The vector W = τ κ is called Darboux vector o the curve α. Then or the Frenet ormulae () we have = W, ( i =,,) ; where " " shows the vector product in M. Deinition.. Let and let ( s) i i M be a -dimensional pseudo-riemannian maniold with the metric g α be a null Frenet curve with the Frenet rame {,, } W = τ κ in M. Let and Darboux vector : M R be an eikonal unction along the curve α, i.e. = constant along α. Then we deine the ollowings, i) I the unction g (, ) is a non-zero constant along α, then α is called a null - eikonal helix curve. And, is called the axis o the null -eikonal helix curve α. ii) I the unction ( ) g,,( i =,) is a non-zero constant along α, then α is called a i null -eikonal i -slant helix curve. And, is called the axis o the null -eikonal slant helix curve α. iii) I the unction g (, W ) eikonal Darboux helix curve. And, curve α. is a non-zero constant along α, then α is called a null - is called the axis o the null -eikonal Darboux helix Example.. We consider the pseudo-riemannnian maniold metric g. Let consider the unction M =R with the Lorentzian : M R (,, ) (,, ) = + + x y z x y z x y z given in Example. and consider the null curve

11 in α : I R M s α( s) = (sinh s,cosh s, s) M. I we compute, we ind out ( ) = + 4 x + y, and, along the curve α, we ind out = 5 = constant. as = ( x, y,). Then, we have That is, is an eikonal unction along α. Moreover, by a simple computation we have that the tangent and binormal o the curve are ( s ) = (cosh s,sinh s,), ( s) = cosh s, sinh s,, respectively. Since ( sinh, cosh, ) = s s,, = = constant which mean that α is both a null -eikonal helix and -eikonal -slant helix in M. along α, we easily see that g (, ) = = constant and g ( ) On the other hand, the curvatures o curve are Darboux vector o α is κ =, τ =, respectively. Then the W = (0, 0, ). Then we obtain that ( W ) g, = = constant, along α. So, α is also a null -eikonal Darboux helix curve in M. Then, we can give the ollowing characterizations or a null curve. Theorem.. Let let R be a null curve in M with non-zero curvatures κ, τ and : M R be an eikonal unction along curve α and the Hessian 0 null -eikonal helix curve in M, then the ollowings hold, H =. I α ( s) is a

12 i) The unction κ is constant. τ τ ii) The axis o null -eikonal helix curve is = c + κ a non-zero constant. Proo. Let α ( s) be a null -eikonal helix curve in smooth unctions a = a ( s), a = a ( s) and a a ( s), where (, ) M with axis = o arc length s such that g = c is. Then, there exist = a + a + a, () where {,, } is a basis o TM (tangent bundle o M ). From () we have (, ) = = = constant, g (, ) = a, (, ) g a c g = a. () Dierentiating equalities given in (), we have a = 0, a = constant and κ a constant τ = a =, respectively. τ Moreover, rom () the axis o the null helix is obtained as = c + κ, where g, = c is a non-zero constant. ( ) Theorem.. Let let α : I R R be a null curve in R with non-zero curvatures κ, τ and H =. I α ( s) is a : R R be an eikonal unction along curve α and the Hessian 0 null -eikonal -slant helix curve in τ R with axis = c + have g ( ) R, then α ( s) is also a null -eikonal helix curve in κ where g (, ) = c is a non-zero constant. Proo: Let α ( s) be a null -eikonal -slant helix curve in R with axis, = non-zero constant. By dierentiation o last equality we get ( ). Then we g, = 0. (4). On the other hand dierentiation o g (, ) ( ), g = κ g(, ), in the direction and rom (4) we have g (, ) is a constant. Then ( s) in R and rom Theorem., the axis is = c + g = c is a nonzero constant. τ κ is α is a null -eikonal helix curve where (, )

13 Theorem.. Let let α : I R R be a null curve in R with non-zero curvatures κ, τ and H =. I α ( s) is a : R R be an eikonal unction along curve α and the Hessian 0 null -eikonal helix or -slant helix in R, then det(,, ) = 0 holds. Proo: Let α ( s) be a null curve. Then rom Frenet ormulae () we have the ollowings = τ τκ + τ, = ττ (( τκ ) + κτ ) + ( κτ + τ ). (5) 5 κ From (5) we have det(,, ) = τ. Then by Theorem. and Theorem τ., we say that i α ( s) is a null -eikonal helix or -slant helix curve in R then det(,, ) = 0 holds. Theorem.4. Let R be a null curve in M with curvatures κ, τ and let : M R be an eikonal unction along curve α and the Hessian 0 -eikonal -slant helix curve in M, then the ollowing properties hold: H =. I α ( s) is a null i) s κ ( s) τ ( s) ds + τ ( s) κ( s) ds = s holds, where (, ) g = c is a non-zero constant. ii) The axis o the -slant helix is given by s s = c τ ( s)ds + κ( s)ds (6) Proo: Since we assume that α ( s) is a null -eikonal -slant helix curve in (, ) g = c is a non-zero constant. Then we can write M, we have = a ( s) + a ( s) + c, (7) where a = a ( s); ( i =,) are the dierentiable unctions o s. From Corollary., is parallel in i i M, i.e., = 0 along α. Then rom (7) we obtain ( a cτ ) + ( a cκ ) + ( a κ + a τ ) = 0, (8) which gives the ollowing system a cτ = 0, a cκ = 0, a κ + a τ = 0. (9) And rom (9) and (7), we have the ollowings immediately,

14 s s = c τ ( s)ds + κ( s)ds s,. κ ( s) τ ( s) ds + τ ( s) κ( s) ds = s Theorem.4 gives us the ollowing corollary: Corollary.. Let R be a null curve in M with curvatures κ, τ and let : M R be an eikonal unction along curve α and the Hessian 0 null -eikonal -slant helix curve in M. Then the ollowings holds, i) α( s) is a null -eikonal helix curve i and only i κ ( s) = 0. ii) α( s) is a null -eikonal -slant helix curve i and only i τ ( s) = 0. Proo: From Theorem.4, we have that the axis is given by = a ( s) + a ( s) + c, where c is a non-zero constant. Then we have that ( ) ( ) H =. I α ( s) is a g, = a ( s), g, = a ( s), (0) and rom (9) and (0) we have the ollowings, i) (, ) g = a is constant i and only i κ ( s) = 0, ii) (, ) g = a is constant i and only i τ ( s) = 0, which inish the proo. Theorem.5. Let W = τ κ in R be a null -eikonal Darboux helix with Darboux vector M with non-zero curvatures κ, τ, where : M R is an eikonal unction along curve α and the Hessian H = 0. Then α is a null -eikonal -slant helix i and only i κτ is constant. Proo. Since α is a null -eikonal Darboux helix, we have g ( W ), = constant. On the other hand, there exist smooth unctions a = a ( s), a = a ( s) and a ( ) = a s o arc length s such that = a + a + a, ()

15 where,, a a a are assumed non-zero and {,, } is a basis o TM. Since along α, i we take the derivative in each part o () in the direction in Frenet equations, we get = 0 M and use the ( a a τ ) + ( a a κ) + ( a + a κ + a τ ) = 0, () where a ( s) = [ a ], ( i =,,). Since the Frenet rame {,, } i we have the system i is linearly independent, a aτ = 0, a aκ = 0, a + aκ + aτ = 0. Equality g ( W ), = constant gives that () aτ aκ = constant. (4) Dierentiating (4) and using the irst and second equations o system () we obtain a τ κ =. (5) a 0 From (5) and the third equation o system () it ollows ( κτ ) a = a. (6) κ I a = 0 in (6), rom () we see that a = a = 0 which is a contradiction. Then a 0 and we have that a ( ) = a s is a constant i and only i κτ is constant, which means that α is a null -eikonal -slant helix i and only i κτ is constant. Reerences [] Ali, A.T., Position vectors o slant helices in Euclidean Space Soc., 0() (0) -6. [] Ali, A.T., Lopez, R., Slant Helices in Minkowski Space E, J. o Egyptian Math. E, J. Korean Math. Soc. 48() (0) [] Ali, A.T., Turgut, M., Position vector o a time-like slant helix in Minkowski -space, J. Math. Anal. Appl. 65 (00) [4] Barros, M., General helices and a theorem o Lancret, Proc. Amer. Math. Soc. 5, no.5, (997) [5] Di Scala, A.J., Ruiz-Hernandez, G., Higher codimensional euclidean helix submaniolds, Kodai Math. J., (00) 9-0. [6] Duggal, K.L., Jin, D.H., Null curves and hypersuraces o semi-riemannian maniolds, World Scientiic, 007. [7] Ekmekçi, N., Hacısalihoğlu, H.H., On Helices o a Lorentzian Maniold, Commun. Fac. Sci. Univ. Ank. Series A, 45 (996) [8] Ferrandez, A., Gimenez, A., Lucas, P., Null generalized helices in Lorentz-Minkowski spaces, J. Phys. A 5, no.9 (00)

16 [9] Ferrandez, A., Gimenez, A., Lucas, P., Null helices in Lorentzian space orms, Internat. J. Modern Phys. A. 6, no.0 (00) [0] Fischer, A.E., Riemannian maps between Riemannian maniolds, Contemporary Math., ol 8, (99) -66. n [] Gök, Đ., Camcı, Ç, Hacısalihoğlu, H.H., n -slant helices in Euclidean n space E, Math. Commun., 4() (009) 7-9. [] Izumiya, S., Takıyama, A., A time-like surace in Minkowski -space which contain pseudo-circles, Proc. Edinburg Math. Soc., 40 (997), 7-6. [] Izumiya, S., Takiyama, A., A time-like surace in Minkowski -space which contain light-like lines, Journal o Geometry, 64 (999), [4] Izumiya, S., Takeuchi, N., New special curves and developable suraces, Turk J. Math. 8, (004) 5-6. [5] Kobayashi, O., Maximal suraces in the -dimensional Minkowski space L, Tokyo J. Math., 6 (98), [6] Kula, L. and Yayli, Y., On slant helix and its spherical indicatrix, Appl. Math. and Comp., 69, (005) [7] Kula, L., Ekmekçi, N., Yaylı, Y., Đlarslan, K., Characterizations o Slant Helices in Euclidean -Space, Turk J Math., (009). [8] O Neill, B., Semi-Riemannian Geometry with Applications to Relativity. Academic Press, London (98). 4 [9] Önder, M., Kazaz, M., Kocayiğit, H. Kılıç, O., B -slant helix in Euclidean 4-space E, Int. J. Contemp. Math. Sci. (9-) (008) [0] Struik, D.J., Lectures on Classical Dierential Geometry, nd ed. Addison Wesley, Dover, (988). [] Şenol, A., Zıplar, E., Yaylı, Y., On -Eikonal Helices and -Eikonal Slant Helices in Riemannian Maniolds, arxiv:.4960 [math.dg]. [] Zıplar, E., Şenol, A., Yaylı, Y., On Darboux helices in Euclidean -space, Global J. o Sci. Frontier Res., () (0) 7-80.