SOOCHOW JOURNAL OF MATHEMATICS Volume 31, No. 3, pp. 441-447, July 2005 THE CHARACTERIZATIONS OF GENERAL HELICES IN THE 3-DIMEMSIONAL PSEUDO-GALILEAN SPACE BY MEHMET BEKTAŞ Abstract. T. Ikawa obtained in [6] the following characteristic ordinary differential equation X X XX K XX = 0, K = k 2 τ 2 for the circular helix which corresponds to the case that the curvatures k and τ of a time-like curve α on the Lorentzian manifold M are constant. N. Ekmekçi and H. H. Hacisalihoglu generalized in [4] T. Ikawa s this result, i.e. k and τ are variable, but k is constant. τ Recently, N. Ekmekçi and K. İlarslan obtained characterizations of timelike null helices in terms of principal normal or binormal vector fields ([5]). In this paper, making use of method in [4, 5, 6] we obtained characterizations of helices in terms of principal normal or binormal vector fields for a curve with respect to the Frenet frame of ruled surfaces in the 3-dimensional pseudo-galilean space G 1 3. 1. Preliminaries The pseudo-galilean geometry is one of the real Cayley-Klein geometries (of projective signature (0,0,+,-)). The absolute of the pseudo-galilean geometry is an ordered triple {w, f, I} where w is the ideal (absolute) plane, f is a line in w and I is the fixed hyperbolic involution of points of f ([1]). A vector X(x, y, z) is said to non isotropic if x 0. All unit non-isotropic vectors are of the form (1, y, z). For isotropic vectors x = 0 holds. There are four types of isotropic vectors: space-like (y 2 z 2 > 0), time-like (y 2 z 2 < 0) and Received June 5, 2004; revised December 10, 2004. AMS Subject Classification. 53B30. Key words. ruled surface, general helix. 441
442 MEHMET BEKTAŞ two types of lightlike (y = ±z) vectors. A non-lightlike isotropic vector is a unit vector if y 2 z 2 = ±1. A trihedron (T o ; e 1, e 2, e 3 ) with a proper origin T o (x o, y o, z o ) (1 : x o : y o : z o ), is orthonormal in pseudo-galilean sense iff the vectors e 1, e 2, e 3 are of following form: e 1 = (1, y 1, z 1 ), e 2 = (0, y 2, z 2 ), e 3 = (0, ɛz 2, ɛy 2 ), with y2 2 z2 2 = δ, where ɛ, δ is +1 or -1. Such trihedron (T o ; e 1, e 2, e 3 ) is called positively oriented if for its vectors det(e 1, e 2, e 3 ) = 1 holds i.e. if y2 2 z2 2 = ɛ. 2. Ruled Surfaces in the Galilean Space A general equation of a ruled surface G 1 3 is x(u, v) = r(u) + va(u), v IR; r, a C 3 (2.1) where the curve r does not line in a pseudo-euclidean plane and is called a directix. The curve r is given by r(u) = (u, y(u), z(u)). (2.2) This means that the curve r is parametrizd by the pseudo-galilean arc length. Further, the generator vector field is of the form a(u) = (0, a 2 (u), a 3 (u)), a 2 2 a 2 3 = 1. (2.3) The equation of a ruled surface in the 3-dimensional pseudo-galilean space G 1 3 is x(u, v) = (u, y(u), 0) + v(0, a 2 (u), a 3 (u)), y, z, a 2, a 3 C 3, u I IR, v IR, a 2 2 (2.4) a2 3 = 1. The associated trihedron of a ruled surface in the 3-dimensional pseudo- Galilean space G 1 3 is defined by T (u) = s (u) = (1, y (u), 0), N(u) = a(u) = (0, a 2 (u), a 3 (u)), (2.5) B(u) = (0, a 3 (u), a 2 (u)).
THE CHARACTERIZATIONS OF GENERAL HELICES 443 The Frenet formulas of a ruled surface in the 3-dimensional pseudo-galilean space G 1 3 is defined by T(u) T (u) 0 k(u)chϕ(u) k(u)shϕ(u) T (u) T(u) N(u) = 0 0 τ(u) N(u) (2.6) T(u) B(u) 0 τ(u) 0 B(u) where k(u) = y (u), τ(u) = a 2 (u) a 3 (u) and ϕ is the angle between a(u) and the plane z = 0 ([2]). B. Divjak and Z. M. Sipus [3] gave an example of ruled surfaces in the 3- dimensional pseudo-galilean space G 1 3 which it can be parametrized as x(u, v) = (bu, chv + b u2, shv), b = const(ir {0}, u, v IR). 2 3. The Characterizations of Curves in the 3-Dimensional Pseudo- Galilean Space Definition 3.1. Let α be a curve of a ruled surface in the 3-dimensional pseudo-galilean space G 1 3 and {T (u), N(u), B(u)} be the Frenet frame on ruled surface along α. If k and τ are positive constants along α, then α is called a circular helix with respect to the Frenet frame. Definition 3.2. Let α be a curve of a ruled surface in the 3-dimensional pseudo-galilean space G 1 3 {T (u), N(u), B(u)} be the Frenet frame on ruled surface along α. A curve α such that k(u) τ(u) = const is called a general helix with respect to Frenet frame. Theorem 3.1. Let α be a curve of a ruled surface in the 3-dimensional pseudo-galilean space G 1 3. α is a general helix with respect to the Frenet frame {T (u), N(u), B(u)} if and only if T(u) T(u) T(u) N(u) K(u) T(u) N(u) 3 λ k (u)τ(u)n(u) = 0, (3.1)
444 MEHMET BEKTAŞ where K(u) = τ (u) τ(u) Proof. + τ 2 (u). Suppose that α is general helix with respect to the Frenet frame {T (u), N(u), B(u)}. Then from (2.6), we have T(u) T(u) T(u) N(u) = (τ (u) + τ 3 (u))b(u) + (3τ(u)τ (u))n(u). (3.2) Now, since α is general helix with respect to the Frenet frame If we substitute the equations and (3.5) in (3.2), we obtain (3.1). k(u) = λ = const. (3.3) τ(u) N(u) = 1 B(u), (3.4) B(u) = 1 N(u) (3.5) Conversely let us assume that the equation (3.1) holds. We show that the curve α is a general helix. Differentiating covariantly (3.5) we obtain T(u) B(u) = τ (u) N(u) + 1 T(u) N(u) (3.6) and so ( T(u) T(u) B(u) = τ (u) τ 2 (u) ) T(u) N(u) 2 τ (u) T(u) N(u) + 1 T(u) T(u) N(u). (3.7) If we use (3.1) in (3.7) and make some calculations, we have [ ( T(u) T(u) B(u) = τ ) (u) τ 2 + K(u) ] T(u) N(u) 2 τ (u) (u) τ(u) T(u) N(u) Also we obtain + 3 λ τ (u)k(u) N(u). (3.8) τ(u) T(u) T(u) B(u) = τ 2 (u)b(u) + τ (u)n(u) (3.9)
THE CHARACTERIZATIONS OF GENERAL HELICES 445 since (3.8) and (3.9) are equal, routine calculations show that α is a general helix. Corollary 3.1. Let α be a curve of ruled surface in the 3-dimensional pseudo-galilean space G 1 3. α is a circular helix with respect to the Frenet frame {T (u), N(u), B(u)}, if and only if T(u) T(u) T(u) N(u) = τ 2 (u) T(u) N(u). (3.10) Proof. From the hypothesis of Corollary 3.1 and since α is a circular helix, we can show easily (3.10). Theorem 3.2. Let α be a curve of a ruled surface in the 3-dimensional pseudo-galilean space G 1 3 α is a general helix with respect to the Frenet frame {T (u), N(u), B(u)} if and only if T(u) T(u) T(u) B(u) K(u) T(u) B(u) = 3 λ k (u) T(u) N(u). (3.11) Proof. Suppose that α is a general helix with respect to the Frenet frame {T (u), N(u), B(u)}. Then from (2.6), we have T(u) T(u) T(u) B(u) = (τ (u) + τ 3 (u))n(u) + (3τ(u)τ (u))b(u). (3.12) (3.3), (3.4) and (3.5) in (3.12), we obtain (3.11). Conversely let us assume that the equation (3.11) holds. We show that the curve α is a general helix. Differentiating covariantly (3.4) we obtain T(u) N(u) = τ (u) B(u) + 1 T(u) B(u) (3.13) and so ( T(u) T(u) N(u) = τ (u) τ 2 (u) ) T(u) B(u) 2 τ (u) T(u) B(u) + 1 T(u) T(u) B(u). (3.14) If we use (3.11) in (3.14) and make some calculations, we have [ ( T(u) T(u) N(u) = τ ) (u) τ 2 + K(u) ] T(u) B(u) 2 τ (u) (u) τ(u) T(u) B(u) +3 τ(u)k (u) B(u). (3.15) k(u)
446 MEHMET BEKTAŞ Also we obtain T(u) T(u) N(u) = τ 2 (u)b(u) + τ (u)b(u) (3.16) since (3.15) and (3.16) are equal, routine calculations show that α is a general helix. Corollary 3.2. Let α be a curve of ruled surface in the 3-dimensional pseudo- Galilean space G 1 3. α is a circular helix with respect to the Frenet frame {T (u), N(u), B(u)}, if and only if T(u) T(u) T(u) B(u) = τ 2 (u) T(u) B(u). (3.17) Proof. From the hypothesis of Corollary 3.2 and since α is a circular helix, we can show easily (3.17). Corollary 3.3. Let α be a curve of ruled surface in the 3-dimensional pseudo- Galilean space G 1 3. If α is a circular helix with respect to the Frenet frame {T (u), N(u), B(u)}, then T(u) T(u) T(u) N(u) T(u) T(u) T(u) B(u) = T N(u) (u) T(u) B(u). (3.18) Proof. From (3.10) and (3.17), we obtain (3.18). Corollary 3.4. Let α be a curve of ruled surface in the 3-dimensional pseudo- Galilean space G 1 3. If α is a circular helix with respect to the Frenet frame {T (u), N(u), B(u)}, then T(u) T(u) T(u) N(u) T(u) T(u) T(u) B(u) = N(u) B(u). (3.19) Proof. From (2.6) and (3.18), we obtain (3.19). References [1] B. Divjak, The general solution of the Frenet system of differential equations for curves in the pseudo-galilean space G 1 3, Math. Com., 2(1997), 143-147. [2] B. Divjak and Z. Milin-Sipus, Special curves on ruled surfaces in Galilean and pseudo- Galilean spaces, Acta Math. Hungar., 98:3(2003), 203-215.
THE CHARACTERIZATIONS OF GENERAL HELICES 447 [3] B. Divjak and Z. Milin-Sipus, Minding isometries of ruled surfaces in pseudo-galilean space, J. Geom., 97:3(2003), 35-47. [4] N. Ekmekçi and H. H. Hacisalihoglu, On helices of a Lorentzian manifold, Commun. Fac. Sci., Üniv. Ank. Series, A1(1996), 45-50. [5] N. Ekmekçi and K. Ilarslan, On characterization of general helices in Lorentzian space, Hadronic J., (To appear). [6] T. Ikawa, On curves and submanifolds in an indefinite-riemannian manifold, Tsukuba J. Math., 9(1985), 353-371. Department of Mathematics, Firat University, 23119 ELAZIĞ / TÜRKİYE. E-mail: mbektas@firat.edu.tr