CHARACTERIZATION OF SLANT HELIX İN GALILEAN AND PSEUDO-GALILEAN SPACES

Transcription

1 SAÜ Fen Edebiyat Dergisi (00-I) CHARACTERIZATION OF SLANT HELIX İN ALILEAN AND PSEUDO-ALILEAN SPACES Murat Kemal KARACAN * and Yılmaz TUNÇER ** *Usak University, Faculty of Sciences and Arts,Department of Mathematics, Eylul Campus,6400,Usak-TURKEY, murat.karacan@usak.edu.tr **Usak University, Faculty of Sciences and Arts,Department of Mathematics, Eylul Campus,6400,Usak-TURKEY,yilmaz.tuncer@usak.edu.tr ABSTRACT We consider a curve ( parameterized by the arc length s in alilean and Pseudo-alilean spaces and denote by T, N, B the Frenet frame of (. We say that is a slant helix if there exists a fixed direction U of and N, U such that the functions N, U and are constant. In this work we give characterizations of slant helices in terms of the curvature and torsion of. ALİLEAN VE PSEUDO-ALİLEAN UZAYLARINDA SLANT HELİSİN KARAKTERİZASYONU ÖZET Bu çalışmada, - boyutlu alilean ve Pseudo alilean uzaylarında yay T, N, B parametreli ve Frenet çatısıyla verilen bir eğrinin, asli normali ile sabit bir doğrultu arasındaki açının sabit olmasını sağlayan slant helis olma durumunu, eğrinin eğrilik ve torsiyonu yardımıyla karakterize ettik. 4

2 SAÜ Fen Edebiyat Dergisi (00-I).INTRODUCTION This definition is motivated by what happens in Euclidean space E. In this setting, we recall that a helix is a curve where the tangent lines make a constant angle with a fixed direction. Helices are characterized by the fact that the ratio is constant along the curve [4,7]. Izumiya and Takeuchi have introduced the concept of Slant helix in Euclidean space by saying that the principal normal lines make a constant angle with a fixed direction [6].They characterize a slant helix if and only if the function (.) is constant. See also [,6,8].Recently, helices in alilean space have been studied depending on the causal character of the curve : see for example [,]. Thus, our definition of slant helix are the alilean and Pseudo-alilean versions of the Euclidean one. Our main results in this work is the following characterization of Slant helices in the spirit of the one given in equation (.). We will assume throughout this work that the curvature and torsion functions do not equal zero..alilean SPACE The alilean space is a three dimensional complex projective space, P, in which the absolute figure w, f, I, I consists of a real plane w (the absolute plane), a real line f w (the absolute line) and two complex conjugate points, I I f, (the absolute point. We shall take, as a real model of the space, a real projective space P, with the absolute w, f consisting of a real plane w and a real 44

3 SAÜ Fen Edebiyat Dergisi (00-I) line f w, on which an elliptic involution has been defined. Let be in homogeneous coordinates w... x : 0 0, f... x 0 0 : 0 : x : x 0 : 0 : x : x. In the nonhomogeneous coordinates, the similarity group H 8 has the form where x a y a z a ij a a a x x a x a cos a sin a sin cos a and are real numbers.for a a subgroup B 6, the group of alilean motions: x a a x 0 y b cx y cos z sin z d ex ysin z cos In, there are four classes of lines: (.),we have have the a) (proper) nonisotropic lines-they do not meet the absolute line f. b) (proper) isotropic lines-lines that do not belong to the plane w but meet the absolute line f. c) unproper nonisotropic lines-all lines of w but f. d) the absolute line f. Planes x constant are Euclidean and so is the plane w. Other planes are isotropic. In what follows, the coefficients a and a a will play a special role. In particular, for a a, (.) defines the group B6 H 8 of isometries of the alilean space. The scalar product in alilean space is defined by X, Y x y, if x 0 or y 0 x y x y, if x 0 and y 0 45

4 SAÜ Fen Edebiyat Dergisi (00-I) where X x, x x and y, y y, Y,. r A curve : I R of the class C r in the alilean space is given defined by ( x) s, y(, z( (.) where s is a alilean invariant and the arc length on.the curvature ( and the torsion ( are defined by det (, (, ( ( y ( z (, ( (.) ( The orthonormal frame in the sense of alilean space ₃ is defined by T (, y(, z( N ( 0, y(, z ( ( ( B 0, z (, y (. ( The vectors (.4) T, N and B in (.4) are called the vectors of the tangent, principal normal and the binormal line of, respectively.they satisfy the following Frenet equations [] T N N B B N. (.5).PSEUDO-ALILEAN SPACE The geometry of the pseudo-alilean space is similar (but not the same) to the alilean space.the pseudo-alilean space is a threedimensional projective space in which the absolute consists of a real plane w (the absolute plane), a real line f w (the absolute line) and a hyperbolic involution on f. Projective transformations which presere the absolute form of a group H 8 and are in nonhomogeneous coordinates can be written in the form 46

5 SAÜ Fen Edebiyat Dergisi (00-I) where x a bx y c dx r y cosh r z sinh z e fx r y sinh r z cosh (.) a, b, c, d, e, f, r and are real numbers. Particularly, for b r, the group (.) becomes the group B6 H8 of isometries (proper motion of the pseudo-alilean space. The motion group leaves invariant the absolute figure and defines the other invariants of this geometry.it has the following form x a x y c dx y cosh z sinh z e fx y sinh z cosh. (.) According to the motion group in the pseudo-alilean space, there are nonisotropic vectors X x y, z, (for which holds x 0 of isotropic vectors: spacelike 0, y z 0 0, y z 0 x ) and four types x, timelike x and two types of lightlike vectors 0, y z.the scalar product of two vectors A a, a a and B b, b b in is defined by,, A, B ab, if a 0 or b 0 ab ab, if a 0 and b 0. (.) A curve ( x(, y(, z( is admissible if it has no inflection points, no isotropic tangents or tangents or normals whose projections on the absolute plane would be light-like vectors.for an admissible curve I R the curvature ( : y ( z( x( and the torsion ( are defined by y( z ( y ( z( (, (. (.4) 5 x( ( 47

6 SAÜ Fen Edebiyat Dergisi (00-I) expressed in components.hence, for an admissible curve I R parameterized by the arc length s with differential : ds dx form, given by ( x, y(, z(, (.5) the formulas (.4) have the following form y ( z ( y ( z ( ( y ( z (, (. (.6) ( The associated trihedron is given by T (, y(, z( N ( 0, y(, z ( ( ( B 0, z (, y (. (, chosen by criterion dett, N, B where, that means (.7). y ( z ( y ( z ( The curve given by (.6) is timelike (resp. spacelike) if N ( is a spacelike(resp. timelike) vector. The principal normal vector or simply normal is spacelike if and timelike if.for derivatives of the tangent (vector) T, the normal N and the binormal B,respectively, the following Serret-Frenet formulas hold T N N B B N. From (.8), we derive an important relation [8], 48 (.8) ( ( N( ( ( B(. (.9)

7 SAÜ Fen Edebiyat Dergisi (00-I) 4.SLANT HELICES IN Definition 4.. A curve is called a slant helix if there exists a constant vector field U in such that the function N is constant. (, U Theorem 4.. Let be a curve parameterized by the arc length s in.then is a slant helix if and only if either one the next two functions (4.) is constant everywhere does not vanish. Proof. Let be a curve in.in order to prove Theorem 4., we first assume that is a slant helix. Let U be the vector field such that the function and a such that N(, U c is constant. There exist smooth functions a U a T( cn( a ( B( ) (4.) ( s As U is constant, a differentiation in (4.) together (4.) gives a 0 a a 0 a c 0. From the second equation in (4.) we have Moreover, if a 0, U, U a constant (4.) a a. (4.4) (4.5) We point out that this constraint, together the second and third equation of (4.) is equivalent to the very system (4.). From (4.4) and (4.5), set a m 49. (4.6)

8 SAÜ Fen Edebiyat Dergisi (00-I) Thus, (4.6) which give a m on I. The third equation in (4.) yields d m c ds on I. This can be written as c. (4.7) m This shows a part of Theorem 4.. Conversely, assume that the condition (4.) is satisfied. In order to simplify the computations, we assume that the function in (4.) is a constant, namely, c.we define U T cn B. (4.8) A differentiation of (4.8) together the Frenet equations in gives du ds 0 that is, U is a constant vector. On the other hand, y ( z ( N(, U c c ( and this means that is a slant helix. a, we obtain, U U c a cons tan t. Then a 0 If 0 from (4.) we have c 0. This means that 0 U contradiction. and 5. SLANT HELICES IN PSEUDO-ALILEAN SPACE Definition 5.. A admissible curve is called a slant helix if there exists a constant vector field U in constant. such that the function N(, U 50 is

9 SAÜ Fen Edebiyat Dergisi (00-I) Theorem 5.. Let be a admissible curve parameterized by the arc length s in two functions.then is a slant helix if and only if either one the next. is constant everywhere does not vanish. Proof. Let be a admissible curve in (5.). In order to prove Theorem 5., we first assume that is a slant helix. Let U be the vector field such that the function N, U c functions a and a such that ( is constant. There exist smooth ( s U a T ( c N( a ( B( ) (5.) As U is constant, a differentiation in (5.) together (5.) gives a 0 a a 0 a c 0. From the second equation in (5.) we have (5.) a a. (5.4) Moreover, if a 0, U, U a constant (5.5) We point out that this constraint, together the second and third equation of (5.) is equivalent to the very system (5.). From (5.4) and (5.5), set Thus, (5.6) which give a m a. m on I. The third equation in (5.) yields (5.6) 5

10 SAÜ Fen Edebiyat Dergisi (00-I) d m c ds on I. This can be written as c. (5.7) m This shows a part of Theorem 5.. Conversely, assume that the condition (5.) is satisfied. In order to simplify the computations, we assume that the function in (5.) is a constant, namely, c.we define U T cn B. (5.8) A differentiation of (5.8) together the Frenet equations in du ds 0 that is, U is a constant vector. On the other hand, y ( z ( N(, U c c ( gives and this means that is a slant helix. a, we obtain, U U c a cons tan t. Then 0 If 0 from (5.) we have c 0. This means that 0 U contradiction. a and 6. REFERENCES []. A. O. Ogrenmis,M.Ergut and M.Bektas,On Helices in the alilean space ₃, Iranian Journal of Science and Technology,Transaction A, Vol, No:A,007 []. A.T.Ali and R.Lopez, On Slant Helices in Minkowski -Space, []. M.Bektas,The Characterizations of eneral Helices in the - Dimensional Pseudo-alilean Space,Soochow Journal of MathematicsVolume, No., pp ,July 005 5

11 SAÜ Fen Edebiyat Dergisi (00-I) [4]. M. do Carmo, Differential eometry of Curves and Surfaces, Prentice Hall,976. [5]. L. Kula, Y. Yayli, On slant helix and its spherical indicatrix, Appl. Math.Comp. 69, , 005. [6]. S. Izumiya, N. Takeuchi, New special curves and developable surfaces, Turk.J. Math ,004 [7]. W. Kuhnel, Differential geometry: Curves, Surfaces, Manifolds. Weisbaden:Braunschweig 999 [8]. Z. Erjavec and B.Divjak,The equiform differential geometry of curves in the pseudo-alilean space,mathematical Communications, -, 008 5