some properties of mannheim curves in galilean and pseudo ...

’iauliai Math. Semin.,7 (15), 2012, 115124SOME PROPERTIES OF MANNHEIM CURVESIN GALILEAN AND PSEUDO-GALILEANSPACESAlper Osman Ö‡RENM “, Handan ÖZTEK N,Mahmut ERGÜTDepartment of Mathematics, Frat University, 23119 Elaz§, Turkey;e-mails: ogrenmisalper@gmail.com, handanoztekin@gmail.com, mergut@rat.edu.trAbstract. The aim of this work is to study the Mannheim curves in3-dimensional Galilean and pseudo-Galilean space. We obtain the characterizationsbetween the curvatures and torsions of the Mannheim partnercurves.Key words and phrases: curvature, Frenet frame, Galilean space, Mannheimcurve, pseudo-Galilean space, torsion.2010 Mathematics Subject Classication: 53A35, 53B30.1. IntroductionIn classical dierential geometry, there are a lot of studies on Bertrand andMannheim curves. Many interesting results on Bertrand and Mannheimcurves have been obtained by many mathematicians (see [2], [7][11], [13]).We can see in most studies, a characteristic property of Bertrand and Mannheimcurves which asserts the existence of a linear relation between curvatureand torsions. There are a lot of works on Bertrand curves but there are rathera few works on Mannheim curves.In this study, we have done a study about some special curves in Galileanand pseudo-Galilean space. However, to the best of author's knowledge,Mannheim curves has not been presented Galilean and pseudo-Galilean spacein depth. Thus, the study is proposed to serve such a need.Our study is organized as follows. In Section 2, some fundamental pro-

116 A. O. Ö§renmi³, H. Öztekin, M. Ergütperties of Galilean and pseudo-Galilean space are given which will be used inthe later sections. In Section 3, we give some characterizations of Mannheimcurves in Galilean space. We also give some properties of Mannheim curvesin pseudo-Galilean space in Section 4.2. PreliminariesThe Galilean space is a three dimensional complex projective space P 3 inwhich the absolute gure {w, f, I 1 , I 2 } consists of a real plane w (the absoluteplane), a real line f ⊂ w (the absolute line) and two complex conjugate pointsI 1 , I 2 ∈ f (the absolute points).We shall take, as a real model of the space G 3 , a real projective spaceP 3 with the absolute {w, f} consisting of a real plane w ⊂ G 3 and a realline f ⊂ w on which an elliptic involution ε has been dened.We introducehomogeneous coordinates in G 3 in such a way that the absolute plane w isgiven by x 0 = 0, the absolute line f by x 0 = x 1 = 0 and elliptic involutionby (x 0 : x 1 : x 2 : x 3 ) = (1 : x : y : z), the distance between the pointsP i = (x i , y i , z i ), i = 1, 2, is dened by{|x 2 − x 1 | if x 1 ≠ x 2 ,d(P 1 , P 2 ) = √(y2 − y 1 ) 2 + (z 2 − z 1 ) 2 if x 1 = x 2 .Let α be a curve in G 3 dened by the arc length α : I → G 3 andparametrized by the invariant parameter s ∈ I, given in the coordinate formα(s) = (s, y(s), z(s)).Then the curvature κ(s) and the torsion τ(s) are dened by√κ(s) = y ′′2 (s) + z ′′2 (s),τ(s) = det(α′ (s), α ′′ (s), α ′′′ (s))κ 2 ,(s)and associated moving trihedron is given byT (s) = α ′ (s) = (1, y ′ (s), z ′ (s)),N(s) =1κ(s) α′′ (s) = 1κ(s) (0, y′′ (s), z ′′ (s)),B(s) =1κ(s) (0, −z′′ (s), y ′′ (s)).

Some properties of Mannheim curves... 117The vectors T, N, B are called the vectors of the tangent, principal normaland binormal line of α, respectively. For their derivatives, the followingFrenet formulas holdT ′ (s) = κ(s)N(s),N ′ (s) = τ(s)B(s),B ′ (s) = −τ(s)N(s).More about the Galilean geometry can be found in [12].The pseudo-Galilean geometry is one of the real Cayley-Klein geometries(of projective signature (0, 0, +, −) explained in [3]. The absolute of thepseudo-Galilean geometry is an ordered triple {w, f, I}, where w is the ideal(absolute) plane, f is the line in w, and I is the xed hyperbolic involutionof points of f.As in [3], pseudo-Galilean scalar product can be written as{x 1 x 2 if x 1 ≠ 0 or x 2 ≠ 0,⟨v 1 , v 2 ⟩ =y 1 y 2 − z 1 z 2 if x 1 = 0 and x 2 = 0,where v 1 = (x 1 , y 1 , z 1 ), v 2 = (x 2 , y 2 , z 2 ).It leaves invariant the pseudo-Galilean norm of the vector v = (x, y, z)dened by{x if x ≠ 0,∥v∥ = √|y 2 − z 2 | if x = 0.A vector v = (x, y, z) is said to be non-isotropic if x ≠ 0. All unit nonisotropicvectors are of the form (1, y, z). For isotropic vectors, x = 0 holds.There are four types of isotropic vectors: spacelike (y 2 − z 2 > 0), timelike(y 2 − z 2 < 0) and two types of lightlike (y = ±z) vectors. A non-lightlikeisotropic vector is a unit vector if y 2 − z 2 = ±1.Let α : I −→ G 1 3 , I ⊂ R, be a curve given byα(t) = (x(t), y(t), z(t)),where x(t), y(t), z(t) ∈ C 3 (the set of three times continuously dierentiablefunctions), and t runs through a real interval [3].A curve α(t) in G 1 3 is said to be admissible curve if x′ (t) ≠ 0 [3].The curves in G 1 3 are characterized as follows [4][5].Type I. Let α be an admissible curve in G 1 3 parametrized by arc lengtht = s, and given in coordinate formα(s) = (s, y(s), z(s)).

118 A. O. Ö§renmi³, H. Öztekin, M. ErgütThen the curvature κ α (s) and the torsion τ α (s) are dened byκ α (s) =√ ∣∣y ′′2 (s) − z ′′2 (s) ∣ ∣,τ α (s) = det(α′ (s), α ′′ (s), α ′′′ (s))κ 2 ,α(s)and the associated moving trihedron is given byT α (s) = α ′ (s) = (1, y ′ (s), z ′ (s))N α (s) =1κ α (s) α′′ (s) = 1κ α (s) (0, y′′ (s), z ′′ (s)),B α (s) =1κ α (s) (0, z′′ (s), y ′′ (s)).The vectors T α , N α , B α are called the vectors of the tangent, principalnormal and binormal line of α, respectively. For their derivatives the followingFrenet formulas holdT α(s) ′ = κ α (s)N α (s),N α(s) ′ = τ α (s)B α (s),B α(s) ′ = τ α (s)N α (s).Type II. Let β be an admissible curve in G 1 3s, and given in coordinate formparametrized by arc lengthβ(s) = (s, y(s), 0).Then the curvature κ β (s) and the torsion τ β (s) are dened byκ β (s) = y ′′ (s),τ β (s) = a′ 2 (s)a 3 (s) ,and the associated moving trihedron is given bywhere, y, a 2 , a 3 ∈ C ∞ , s ∈ I ⊆ R.T β (s) = (1, y ′ (s), 0),N β (s) = (0, a 2 (s), a 3 (s)),B β (s) = (0, a 3 (s), a 2 (s)),

Some properties of Mannheim curves... 119The vectors T β , N β , B β are called the vectors of the tangent, principalnormal and binormal line of β, respectively. For their derivatives, the followingFrenet formulas holdT β ′ (s) = κ β(s)(cosh ϕ(s)N β (s) − sinh ϕ(s)B β (s)),N β ′ (s) = τ β(s)B β (s),B β ′ (s) = τ β(s)N β (s), (2.1)where ϕ(s) is the angle between a(s) = (0, a 2 (s), a 3 (s)) and the plane z = 0.3. Mannheim curves in Galilean spaceDefinition 3.1. Let α and α 1 be curves in 3-dimensional Galilean space G 3 .If there exists a corresponding relationship between the space curves α andα 1 such that, at the corresponding points of the curves, the principal normallines of α coincide with the binormal lines of α 1 , then α is called a Mannheimcurve and α 1 is called a Mannheim partner curve of α. The pair {α, α 1 } issaid to be a Mannheim pair [6].Remark 3.2. Let α be a curve in Galilean space G 3 . Then α is Mannheimcurve if and only if its curvature κ α and torsion τ α satisfy the relation κ α =cτ 2 α for some constant c [10].Theorem 3.3. Let α be a Mannheim curve in Galilean space G 3 . Then α 1is the Mannheim partner curve of α. Then the curvature κ 1 and the torsionτ 1 of α 1 satisfy the equationfor some nonzero constant λ.τ ′ 1 = κ 1λ (λ2 τ 2 1 + 1)Proof. Suppose that α(s) is a Mannheim curve in Galilean space G 3 . Thenwe can writeα(s 1 ) = α 1 (s 1 ) + λ(s 1 )B 1 (s 1 ) (3.1)for some function λ(s 1 ). By taking derivative of (3.1) with respect to s 1 andusing the Frenet equations in Galilean space G 3 , we getT dsds 1= T 1 + λ ′ B 1 − λτ 1 N 1 .Since B 1 is coincident with N, we obtainλ ′ (s 1 ) = 0,

120 A. O. Ö§renmi³, H. Öztekin, M. Ergütthat means that λ is nonzero constant. Thus, we haveOn the other hand, we haveT dsds 1= T 1 − λτ 1 N 1 . (3.2)T = T 1 cos θ + N 1 sin θ, (3.3)where θ is the angle between T and T 1 at the corresponding points of α andα 1 . Dierentiating of (3.3) with respect to s 1 , we getκN dsds 1= −T 1 θ ′ sin θ + κ 1 N 1 cos θ + N 1 θ ′ cos θ + τ 1 B 1 sin θ,κN dsds 1= −T 1 θ ′ sin θ + N 1 (κ 1 + θ ′ ) cos θ + τ 1 B 1 sin θ.Since {α, α 1 } is a Mannheim pair, we obtainκ 1 + θ ′ = 0,and, therefore, we haveθ ′ = −κ 1 . (3.4)From (3.2) and (3.3), we nd thatλτ 1 = − tan θ. (3.5)By taking derivative of this last equation and applying (3.4), we obtainIf we consider (3.4) and (3.5) in (3.6), we getHence, the proof is completed.λτ ′ 1 = −θ ′ (1 + tan 2 θ). (3.6)τ ′ 1 = κ 1λ (λ2 τ 2 1 + 1).Proposition 3.4. Let α be a Mannheim curve in Galilean space G 3 and α 1be the Mannheim partner curve of α. If α is a generalized helix, then α 1 is aplanar curve.

Some properties of Mannheim curves... 121Proof. Let T, N, B be the tangent, principal normal and binormal vectoreld of the curve α, respectively. From the properties of generalized helixand the denition of Mannheim curves in Galilean space G 3 , we haveandκ < N, P >= 0κ < B 1 , P >= 0for a constant direction P in Galilean space G 3 . Then it is easy to obtainthat τ 1 = 0.4. Mannheim curves in pseudo-Galilean spaceDefinition 4.1. Let α and α 1 be an admissible curves dened in Type Iwith nonzero κ α , τ α , s ∈ I in pseudo-Galilean space and {T α , N α , B α } and{T α1 , N α1 , B α1 } be the Frenet frame in pseudo-Galilean space G 1 3 along αand α 1 , respectively. If there exists a corresponding relations between theadmissible curves α and α 1 such that, at the corresponding points of theadmissible curves, principal normal lines N α of α coincides with the binormallines B α1 of α 1 , then α is called an admissible Mannheim curves and α 1 iscalled an admissible Mannheim partner curve of α. The pair {α, α 1 } is saidto be an admissible Mannheim pair in pseudo-Galilean space G 1 3 [1].Theorem 4.2. Let α an admissible curve dened in Type I in pseudo-Galilean space G 1 3 . Then α is an admissible Mannheim curve if and onlyif its curvature κ α and torsion τ α satisfy the relation κ α = −cτα 2 for someconstant c.Proof. Let α = α(s) be an admissible Mannheim curve. Let us denote by{T α , N α , B α } the Frenet frame eld of α.Assume that α 1 = α 1 (s 1 ) is an admissible curve whose binormal directioncoincides with the principal normal of α. Namely let us denote by{T α1 , N α1 , B α1 } the Frenet frame eld of α 1 . Then B α1 (s 1 ) = ± N α (s).The curve α 1 is parametrized by arc length s asα 1 (s) = α(s) + λ(s)N α (s) (4.1)for some function λ(s) ≠ 0. Dierentiating (4.1) with respect to s, we ndα ′ 1 = T α + λ ′ N α + λτ α B α .

122 A. O. Ö§renmi³, H. Öztekin, M. ErgütSince the binormal direction of α 1 coincides with the principal normal of α,we have λ ′ = 0. Hence, λ is constant. The second derivative α 1 ′′ with respectto s isα 1 ′′ = (κ α + λτα)N 2 α + λτ αB ′ α .Since N α is in the binormal direction of α 1 , we haveκ α + λτ 2 α = 0.Conversely, let α be an admissible curve. Then the curvehas binormal direction N α .α 1 (s) = α(s) + λN α (s)Theorem 4.3. Let β an admissible curve dened in Type II in pseudo-Galilean space G 1 3 . Then β is an admissible Mannheim curve if and only ifits curvature κ β and torsion τ β satisfy the relation κ β =−ccosh ϕ τ β 2 for someconstant c.Proof. If we consider equations (2.1) and proof of the Theorem 4.2. wecan prove the theorem easily.Theorem 4.4. Let α be an admissible Mannheim curve dened by Type Iin pseudo-Galilean space G 1 3 . Then α 1 is the admissible Mannheim partnercurve of α. Then the curvature κ 1 and the torsion τ 1 of α 1 satisfy the followingequationτ ′ 1 = κ 1λ (λ2 τ 2 1 − 1)for some nonzero constant λ.Proof. It is similar to a proof of Theorem 3.3.Remark 4.5. By a simple parameter transformation, the conditioncan be written asτ ′ 1 = κ 1λ (λ2 τ 2 1 − 1)τ 1 = − ε ∫ (ελ tanκ 1 ds + c 0).Therefore, for each admissible Mannheim curve in pseudo-Galilean space G 1 3 ,there is an unique Mannheim partner curve.This reality is true for Mannheim curve in Galilean space G 3 .

Some properties of Mannheim curves... 123Proposition 4.6. Let α be an admissible Mannheim curve in pseudo-Galileanspace G 1 3 and α 1 be the admissible Mannheim partner curve of α. If α is ageneralized helix, then α 1 is a planar curve.Proof. Let T, N, B be the tangent, principal normal and binormal vectoreld of the curve α, respectively. From the properties of generalized helixand the denition of admissible Mannheim curves in pseudo-Galilean spaceG 1 3 , we have κ < N, P >= 0andκ < B 1 , P >= 0for a constant direction P in pseudo-Galilean space G 1 3 . Then it is easy toobtain that τ 1 = 0.References[1] M. Akyi§it, A. Z. Azak, M. Tosun, Admissible Mannheim curves in pseudo-Galilean space G 1 3, African Diaspora J. Math., 10(2), 5865 (2010).[2] M. P. do Carmo, Dierential Geometry of Curves and Surfaces, Pearson Education,N. J., 1976.[3] B. Divjak, Curves in pseudo-Galilean geometry, Ann. Univ. Sci. Budapest.Eötvös, Sect. Math., 41, 117128 (1998).[4] B. Divjak, Z. M. Sipus, Special curves on ruled surfaces in Galilean andpseudo-Galilean spaces, Acta Math. Hungar., 98(3), 203215 (2003).[5] B. Divjak, Z. M. Sipus, Minding isometries of ruled surfaces in pseudo-Galilean space, J. Geometry, 77, 3547 (2003).[6] S. Ersoy, M. Akyi§it, M. Tosun, A note on admissible Mannheim curves inGalilean space G 3 , Math. Combin., Book Ser., 1, 8893 (2011).[7] M. Külahc, M. Ergüt, Bertrand curves of AW (k)-type in Lorentzian space,Nonlinear Anal., 70(4), 17251731 (2009).[8] H. Liu, F. Wang, Mannheim partner curves in 3-space, J. Geometry, 88,120126 (2008).[9] K. Orbay, E. Kasap, On Mannheim partner curves in E 3 , Intern. J. of PhysicalScien., 4(5), 261264 (2009).

124 A. O. Ö§renmi³, H. Öztekin, M. Ergüt[10] A. O. Ö§renmi³, H. Öztekin, M. Ergüt, Bertrand curves in Galilean spaceand their characterizations, Kragujevac J. Math., 32, 139147 (2009).[11] H. B. Öztekin, M. Ergüt, Null Mannheim curves in the Minkowski 3-spaceE13, Turk. J. Math., 34, 18 (2010).[12] O. Röschel, Die Geometrie des Galileischen Raumes, Habilitionsschrift,Leoben, 1984.[13] M. Yildirim Yilmaz, M. Bekta³, General properties of Bertrand curves inRiemann-Otsuki space, Nonlinear Anal., 69(10), 32253231 (2008).Received1 November 2011