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SURFACES OF REVOLUTION WITH POINTWISE 1-TYPE GAUSS MAP IN PSEUDO-GALILEAN SPACE

In this paper, we study surfaces of revolution in the three dimensional pseudo-Galilean space. We classify surfaces of revolution generated by a non-isotropic curve in terms of the Gauss map and the Laplacian of the surface. Furthermore, we give the classification of surfaces of revolution generated by an isotropic curve satisfying pointwise 1-type Gauss map equation.


Introduction
In late 1970's B.-Y. Chen introduced the notion of Euclidean immersions of finite type. Essentially these are submanifolds whose immersion into the mdimensional Euclidean space E m is constructed by making use of a finite number of E m -valued eigenfunctions of their Laplacian. The first results on this subject have been collected in the book [2]. Many works were done to characterize or classify submanifolds in terms of finite type. In a framework of the theory of finite type, B.-Y. Chen and P. Piccini [4] made a general study on submanifolds of Euclidean spaces with finite type Gauss map. Several geometers also studied submanifolds of Euclidean spaces or pseudo-Euclidean spaces with finite type Gauss map.
From the above definition one can see that a submanifold has 1-type Gauss map G if and only if G satisfies the equation for a constant λ and a constant vector C, where ∆ denotes the Laplace operator on a submanifold. A plane, a circular cylinder and a sphere are surfaces with 1-type Gauss map.
Similarly, a submanifold is said to have pointwise 1-type Gauss map if the Laplacian of its Gauss map takes the form for a non-zero smooth function F and a constant vector C. More precisely, a pointwise 1-type Gauss map is said to be of the first kind if (1.2) is satisfied for C = 0, and of the second kind if C = 0. A helicoid, a catenoid and a right cone are the typical examples of surfaces with pointwise 1-type Gauss map. Many results of submanifolds with pointwise 1-type Gauss map were obtained in [1], [3], [5], [6], [7], [9], [12], etc, when the ambient spaces are the Euclidean space, Minkowski space and Galilean space.
In this paper, we study surfaces of revolution in the three dimensional pseudo-Galilean space G 1 3 in terms of their Gauss map. In Sections 2 and 3, we introduce pseudo-Galilean space and construct surfaces of revolution in G 1 3 by non-isotropic and isotropic rotations. In Section 4, we obtain the complete classification of surfaces of revolution generated by non-isotropic curve with pointwise 1-type Gauss map. In the last section, we focus on surfaces of revolution generated by isotropic curve with pointwise 1-type Gauss map and give the complete classification of such surfaces.

Pseudo-Galilean space
Let us recall the basic facts about the three dimensional pseudo-Galilean space G 1 3 . The geometry of the pseudo-Galilean space has been firstly explained in [10]. The pseudo-Galilean space G 1 3 is a Cayley-Klein space with the absolute figure consisting of an ordered triple {ω, f, I}, where ω is the ideal (absolute) plane in the three dimensional real projective space RP 3 , f the line (the absolute line) in ω and I the fixed hyperbolic involution of points of f . Homogenous coordinates in G 1 3 are introduced in such a way that the absolute plane ω is given by x 0 = 0, the absolute line f by x 0 = x 1 = 0 and the hyperbolic involution η by η : (x 0 : x 1 : x 2 : x 3 ) → (0 : 0 : x 3 : x 2 ). Let x = (x 1 , y 1 , z 1 ) and y = (x 2 , y 2 , z 2 ) be two vectors in G 1 3 . A vector x is called isotropic if x 1 = 0, otherwise it is called non-isotropic. The pseudo-Galilean scalar product of x and y is defined by From this, the pseudo-Galilean norm of a vector x in G 1 3 is given by ||x|| = | x, x | and all unit non-isotropic vectors are the form (1, y 1 , z 1 ). There are four types of isotropic vectors: spacelike (y 2 1 − z 2 1 > 0), timelike (y 2 1 − z 2 1 < 0) and the two types of lightlike (y 1 = ±z 1 ) vectors. A non-lightlike isotropic vector is a unit vector if y 2 1 − z 2 1 = ±1.
Let us denote g i = ∂x ∂ui , h ij = ∂x ∂ui , ∂x ∂uj (i, j = 1, 2), where ∼ stands for the projection of a vector on the pseudo-Euclidean yz-plane. A surface M is called admissible if it does not have Euclidean tangent planes. Therefore a surface M is admissible if and only if x ,i = 0 for some i = 1, 2.
Let M be an admissible surface in G 1 3 . Then, the corresponding matrix of the first fundamental form ds 2 of a surface M is given by (cf. [11]) . In such case, we denote the coefficients of ds 2 by g * ij . On the other hand, the unit normal vector field U of a surface M is defined by The Gaussian curvature K of a surface M is defined by means of the coefficients L ij (i, j = 1, 2) of the second fundamental form, which are the normal components of x ,i,j (i, j = 1, 2), that is, Thus, the Gaussian curvature K of M is defined by W 2 and the mean curvature H is given by where ǫ(= ±1) is the sign of the unit normal vector field.
For the coefficients g * ij of the first fundamental form on M we denote by (g * ij ) the inverse matrix of the matrix (g * ij ). In terms of a local coordinate system {x i }, the Laplacian ∆f of a smooth function f is given by where g denotes the determinant of the matrix (g * ij ).

Surfaces of revolution in
In the three dimensional pseudo-Galilean space G 1 3 , there are two types of rotations: pseudo-Euclidean rotations given by the normal form and isotropic rotations with the normal form where t ∈ R and b is a positive constant. First of all, we consider a non-isotropic curve α parameterized by around the x-axis by pseudo-Euclidean rotation (3.1), where g is a positive function and f is a smooth function on an open interval I. Then the surface of revolution can be written as for any v ∈ R. Next, we consider the isotropic rotations. By an isotropic curve α(u) = (0, f (u), g(u)) about the z-axis by an isotropic rotation (3.2), we obtain a surface where f and g are smooth functions and b = 0 [11].

Surfaces of revolution generated by non-isotropic curve
Let M be a surface of revolution generated by non-isotropic curve α(u) where g is a positive function. The coefficients of the first fundamental form on M are given by We see that M is a time-like surface. By a direct computation with the help of (2.5), the Laplacian ∆ on M is given by [12] (4.2) Also, the Gauss map G of M becomes    Remark. We consider a surface defined by where a, b ∈ R and u > − b 2 a 2 . The surface is a Lorentzian cone satisfying the equation (a 2 x + b 2 ) 2 = y 2 − z 2 . From (4.4) the Laplacian ∆G of the Gauss map G of the surface is obtained by ∆G = 1 (a 2 u+b 2 ) 2 G. Thus, a Lorentzian cone in G 1 3 has pointwise 1-type Gauss map of the first kind. On the other hand, a Lorentzian cone in the three dimensional Minkowski space E 3 1 has pointwise 1-type Gauss map of the second kind (see [8]).

Surfaces of revolution generated by isotropic curve
In this section, we consider the isotropic rotations. By rotating an isotropic curve α(u) = (0, f (u), g(u)) about the z-axis, we obtain a surface of revolution M parameterized by where b is a non-zero constant. We assume that the isotropic curve is parameterized by arc-length, that is, By using (5.2), the coefficients of the first fundamental form ds 2 on M are given by On the other hand, the Gauss map G and the Laplacian ∆ on M , respectively, are given by Hence the Laplacian ∆G of the Gauss map G is obtained by [12] (5.4) ∆G = (0, −ǫg ′′′ , −ǫf ′′′ ).
In terms of the harmonic Gauss map, we have:

Surfaces of revolution with pointwise 1-type Gauss map of the first kind
Let M be a surface of revolution in G 1 3 satisfying ∆G = F G for a non-zero smooth function F . Then from (5.4) we have If f ′ = 0, from (5.2) g(u) = ±u + a with a ∈ R and from the first equation of (5.5) F = 0, a contradiction. Similarly, the case of g ′ = 0 is also impossible. Now we suppose that f ′ g ′ = 0. If ǫ = −1, then we put where θ is a smooth function. By substituting (5.6) into (5.5) and calculating we obtain θ ′′ = 0, that is, θ(u) = a 1 u + a 2 , a 1 , a 2 ∈ R. From this and (5.5) we can show that F = −a 2 1 . Therefore, the Gauss map G of M is of 1-type. In the case of ǫ = 1 we have the same result.
Theorem 5.2 (The Classification Theorem). Let M be a surface of revolution generated by an isotropic curve in the three dimensional Galilean space G 1 3 . If M has pointwise 1-type Gauss map of the first kind, then the Gauss map of M is of usual 1-type. Furthermore, M is parameterized as where a i , d i ∈ R, i = 1, 2, 3.

Surfaces of revolution with pointwise 1-type Gauss map of the second kind
Suppose that M has pointwise 1-type Gauss map of the second kind, that is, M satisfies ∆G = F (G + C). Then we easily see that the first component c 1 of a constant vector C = (c 1 , c 2 , c 3 ) is zero and we have a system of differential equations as follows: If f ′ = 0, then ǫ = 1 and g(u) = ±u + k 1 , where k 1 is constant. Hence from (5.7) we have F (±1 + c 2 ) = 0 and F c 3 = 0.
Thus, M is parameterized by where k 1 and k 2 are constant. The surface is a time-like parabolic cylinder and it has pointwise 1-type Gauss map of the second kind. Similarly as above, if g ′ = 0, then f (u) = ±u + k 1 for some constant k 1 and so ǫ = −1. Hence, C = (0, 0, ±1) and moreover C = −G. Therefore M has pointwise 1-type Gauss map of the second kind and its parametrization is given by for some constants k 1 and k 2 . We see that it is a space-like plane. Suppose that f ′ g ′ = 0. In this case, we first consider ǫ = −1 and put f ′ (u) = cosh θ(u) and g ′ (u) = sinh θ(u).
If θ ′ = 0 identically, then f ′ and g ′ are constants, say a 1 and b 1 . It follows that we have ∆G = 0 and C = −G. Thus M has pointwise 1-type Gauss map of the second kind and it is parameterized by for some constants a 1 , a 2 , b 1 and b 2 .
Next, suppose that θ ′ = 0. From (5.8), F depends only on the parameter u, i.e., F (u, v) = F (u). Differentiating equation (5.8) with respect to u, we get By using (5.9), it implies that Combining (5.8) and (5.10), we have the following equation for a non-zero constant k.
Applying the composition of trigonometric function in (5.8) and (5.9), we obtain the following differential equation With the help of (5.11), it becomes Without loss of generality, we assume that k = ±1. In order to solve the above equation, we put p = ln F. Then, (5.12) can be rewritten as the following equation: Let us distinguish three cases according to the constant vector C.
In this case, we can easily obtain a general solution given by where d 1 is non-zero constant. Therefore, from (5.11) we have where d 2 is constant. Thus, M is parameterized as Case 2. C is time-like, that is, c 2 2 − c 2 3 < 0. We assume that c 2 2 − c 2 3 = −1. Then (5.13) becomes dp du 2 = (e p + 1) 2 − 1 and its general solution is given by where d 1 is constant. Thus a first integration of (5.11) implies where d 2 is constant. Thus, the parametrization of M is given by Case 3. C is space-like, that is, c 2 2 − c 2 3 > 0. In the case, we assume that c 2 2 − c 2 3 = 1. (5.13) becomes (5.14) dp du = ± (e p + 1) 2 + 1.
The last equation can be written as the form: where d 1 is constant. This implies From here, we have two values for F . First, by taking the sign + and using (5.11) with k = ±1 we get Finally, by taking the sign − we get where d 2 is constant. In this case, the parametrization of M is given by where θ(u) is given by (5.17) or (5.18). Consequently, we have Theorem 5.3 (The Classification Theorem). Let M be a surface of revolution generated by isotropic curve in the three dimensional Galilean space G 1 3 . If M has pointwise 1-type Gauss map of the second kind, then M is one of the following surfaces: 2b , sinh θ(u)du , where θ(u) is given by (5.17) or (5.18).