On the Gauss Map of Ruled Surfaces of Type II in 3-Dimensional Pseudo-Galilean Space

The classification of submanifold in Euclidean or Non-Euclidean spaces has been of particular interest for Geometers. In 1983, the authors classified spacelike ruled surfaces in a three-dimensional Minkowski space R31 [13], and De Woestijne [11] extended it to the Lorentz version in 1998. In late 1970s Chen [4,5] introduced the notion of Euclidean immersions of finite type. In a framework of theory of finite type, Chen and Piccinni [6] characterized the submanifold satisfying the condition ∆G = λG (λ ∈ R), where ∆ in the Laplacian of the induced metric and G the Gauss map for submanifold. Submanifold of Euclidean and pseudo-Euclidean spaces with finite type Gauss map also studied by following geometers. (cf. [7,9,12,15,16], etc.) On the otherhand, for the Gauss map of a surface in a three-dimensional Euclidean space R the following theorem is proved by Boikoussis and Blair [8].


Introduction
The classification of submanifold in Euclidean or Non-Euclidean spaces has been of particular interest for Geometers.In 1983, the authors classified spacelike ruled surfaces in a three-dimensional Minkowski space R 3  1 [13], and De Woestijne [11] extended it to the Lorentz version in 1998.In late 1970s Chen [4,5] introduced the notion of Euclidean immersions of finite type.In a framework of theory of finite type, Chen and Piccinni [6] characterized the submanifold satisfying the condition ∆G = λG (λ ∈ R), where ∆ in the Laplacian of the induced metric and G the Gauss map for submanifold.Submanifold of Euclidean and pseudo-Euclidean spaces with finite type Gauss map also studied by following geometers.(cf.[7,9,12,15,16], etc.) On the otherhand, for the Gauss map of a surface in a three-dimensional Euclidean space R 3 the following theorem is proved by Boikoussis and Blair [8].
Theorem 1.1.The only ruled surfaces in R 3 whose Gauss map ξ satisfies are locally the plane and the circular cylinder.
Also, for the Lorentz version Choi [15] investigated ruled surfaces with non-null base curve satisfying the condition (1.1) in a three-dimensional Minkowski space R 3  1 .

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Alper Osman Öğrenmiş and Mahmut Ergüt It seems to be interesting to investigate the Pseudo-Galilean version of the above theorem.Now, let G 1 3 be a three-dimensional Pseudo-Galilean space with standard coordinate system {X A }. Let S 1 m (resp.H m ) be an m-dimensional de Sitter space (resp.a hyperbolic space) in a m + 1−dimensional Pseudo-Galilean space G 1 m+1 .We denote by X m (ε) a de Sitter space S 1 m (1) or a hyperbolic space H m (−1), according as ε = 1 or ε = −1.Let X be a surface in G 1  3 and ξ be a unit vector field normal to X.Then, for any point ρ in X, we can regard ξ(ρ) as a point in H 2 (−1) or S 1 2 (1) by translating parallelly to the origin in ambient space G 1  3 , according as ε = ∓1.The map ξ of X into X 2 (ε) is called a Gauss map of X in G 1  3 .Then we prove the following Theorem 1.2.The only ruled surfaces of type II in G 1 3 whose Gauss map ξ : X → X 2 (ε) satisfies (1.1) are locally the following spaces: The theorem is proved in section 3.

Ruled Surfaces
First of all, we recall fundamental properties in three-dimensional Pseudo-Galilean space.
Differential geometry of the Galilean space G 3 has been largely developed in O. Röschel's paper [14].
The Pseudo-Galilean geometry is one of the real Cayley-Klein geometries (of projective signature (0, 0, +, −), explained in [10].The absolute of the Pseudo-Galilean geometry is an ordered triple {w, f, I} where w is the ideal (absolute) plane, f is line in w and I is the fixed hyperbolic involution of points of f.
The group of motions of G 1 3 is a six-parameter group given (in affine coordinates) by As in [1], Pseudo-Galilean scalar product g can be written as where It leaves invariant the Pseudo-Galilean norm of the vector v = (x, y, z) defined by v = x, x = 0 (2.2) On the Gauss Map of Ruled Surfaces of Type II
A trihedron (T 0 ; e 1 , e 2 , e 3 ), with a proper origin is orthonormal in Pseudo-Galilean sense if the vectors e 1 , e 2 , e 3 are of the following form: given by its parametrization where r is the directrix (parametrized by the Pseudo-Galilean arc length) and a is a unit generator vector field.according to the position of the striction curve with respect to the absolute figure, we have three types of skew ruled surfaces in G 1 3 .Throughout this paper, we assume that all objects are smooth unless otherwise mentioned.Now, we define a ruled surface of type II in G 1  3 .A ruled surface of type II in G 1  3 is a surface parametrized by It is a surface whose striction curve r(u) = (0, y(u), z(u)) lies in pseudo-Euclidean plane, u is the arc-lenght on the striction curve, and whose generators a(u) = (1, a 2 (u), a 3 (u)) are non-isotropic.In particular, if a(u) is constant, then it said to be cylindrical, and if it is not so, then the surface is said to be noncylindrical.Since our discussion is local, we may assume that we always have a ′ (u) = 0 in the non-cylindrical case.That is, the direction of the rulings is always changing.[3] The natural basis {X u , X v } along the coordinate curves are given by Accordingly we see For the components g ij of the Pseudo-Galilean metric g we denote (g ij ) (resp.g) the inverse matrix (resp.the determinant) of matrix (g ij ).Then the Laplacian ∆ is given by (2.5)

Cylindrical Ruled Surfaces
In this section we are concerned with cylindrical ruled surfaces.Let X be a cylindrical ruled surface swept out by the vector field a along the base curve r in G 1  3 .That is, r = r(u) is isotropic curve and a = a(u) is a non-isotropic unit constant vector along orthogonal to r.Then the cylindrical ruled surface X is only or type II.And X is parametrized by Then we get g(ξ, ξ) = ε (= ∓1).Let X 2 (ε) be a two-dimensional space form as follows: Then, for any point x in X, ξ(x) can be regarded as a point in X 2 (ε) and the map ξ : X → X 2 (ε) is the Gauss map of X into X 2 (ε).
We give here theorem of ruled surface of type II whose Gauss map satisfies Proof: Let X be a cylindrical ruled surface of type II in G 1  3 parametrized by On the Gauss Map of Ruled Surfaces of Type II 149 where a is a unit constant vector along the curve r orthogonal to it.That is, it satisfies g(r ′ , a) = 0, g(a, a) = 1.Acting a Pseudo-Galilean transformation, we may assume that a = (1, 0, 0) without loss of generality.Then r may be regarded as the plane curve r(u) = (0, y(u), z(u)) parametrized by arc length: The Gauss map ξ is given by ξ = (0, εz ′ (u), εy ′ (u)).It is unit normal to X. Since the induced Pseudo-Galilean metric g is given by g 11 = ε, g 12 = 0 and g 22 = 1, the Laplacian of ξ is given by ∆ξ = (0, εz ′′′ , εy ′′′ ) from (2.5).Thus, from the condition (3.1) we have the following system of differential equation: where A = (a ij ) is the constant matrix.Now, in order to prove this theorem we may solve this equation and obtain the solution y and z.First we consider that ε = −1.So we get g(r Accordingly we can parametrize as follows: where θ = θ(u).Differentiating (3.5), we obtain because sinh θ cosh θ, sinh 2 θ and cosh 2 θ are linearly independent functions of θ = θ(u).Combining the above equations with (3.8) gives where − 1 r 2 = a 22 = a 33 .Accordingly we have This representation gives us to We denote by S 1 1 (r,(c 1 , c 2 )) the pseudo-circle centered at (c 1 , c 2 ) with radius r in the Pseudo-Galilean plane G 1 2 (the yz−plane).By the above equation the curve r is contained in S 1 1 (r,(c 1 , c 2 )) and hence the ruled surface X is contained in the Pseudo-Galilean circular cylinder S 1 1 × R. Now, we consider that ε = 1.So we get g(r ′ , r ′ ) = y ′ 2 − z ′ 2 = 1.Accordingly we can parametrize as follows:

. 1 ) 3 . 1 .
Theorem The only cylindrical ruled surface of type II in G1  3 whose Gauss map satisfies the condition (3.1) are locally the plane and the hyperbolic cylinder (resp.the Pseudo-Galilean plane and Pseudo-Galilean circular cylinder).