Polynomial Affine Translation Surfaces in Euclidean 3-Space

In this paper we study the polynomial affine translation surfaces in E3 with constant curvature. We derive some non-existence results for such surfaces. Several examples are also given by figures.


Introduction
A surface in Euclidean 3-space is called a translation surface if it is the graph surface of the function z (x, y) = f (x) + g (y) , where f and g are smooth functions.Such surfaces are obtained by translating two planar curves.This class of the surfaces are well-studied classical surfaces in Euclidean and Lorentzian space [1,2,3,6,9].A polynomial translation surface [8,10] is parametrized by r : U ⊆ E 2 → E 3 , (x, y) → r (x, y) = (x, y, f (x) + g (y)) , where f and g are polynomial functions on U .
Most recently H. Liu and Y. Yu introduced a new translation surfaces so-called affine translation surfaces.The affine translation surface in Euclidean 3-space is defined as a parameter surface r (u, v) in E 3 which can be written as r (u, v) = (u, v, f (u) + g (v + au)) , for some non zero constant a and smooth functions f (u) and g (v + au) .

196
H. Gün Bozok and M. Ergüt with constant sectional curvature in 4-dimensional affine space, by proving that such surfaces must be flat and one of the defining curves must be planar.Affine translation surfaces with constant Gaussian curvature in 3-dimensional affine space are investigated by Y. Fu and Z. Hou and they obtained a complete classification of such surfaces [4].Also, Y. Yuan and H. L. Liu dealth with translation surfaces of some new types in 3-Minkowski space [11].
In this paper we investigate the affine translation surfaces with constant curvature in E 3 , then we provided non-existence theorems for these surfaces.

Polynomial Affine Translation Surface with Constant Gaussian and Mean Curvature
Let , denote the standart scalar product on E 3 and let .be the induced norm.Consider the affine translation surface M in E 3 parametrized by where f and g are real-valued and smooth functions on U and a is a non-zero constant.Then the first fundamental form of M can be written as where and here f ′ = df (x) dx and g ′ = dg (v) dv = dg (y + ax) d (y + ax) for v = y + ax.The unit normal vector field so-called the Gauss map of M is given by Then the second fundamental form of M is where and here Hence the Gauss and mean curvatures of M are given, respectively, Polynomial Affine Translation Surfaces in Euclidean 3-Space 197 and Note that the affine translation surface given by (2.1) is flat, i.e.K ≡ 0, if and only if at least one of f or g is a linear function.
Example 2.1.Let M be an affine translation surface in E 3 parametrized by It is easy to see that M is a parabolic cylinder and flat.It can be plotted as in Fig. 1.  = (x, y, z (x, y)) be a minimal affine translation surface.Then either z (x, y) is linear or can be written as . (2.4) The minimal translation surface given by (2.4) is called generalized Sherk surface or affine Sherk surface in E 3 .We plot it as in Fig. 2. Now, we consider the polynomial affine translation surfaces parametrized by where f and g are polynomial functions on U .Therefore, the following nonexistence results for polynomial affine translation surfaces can be provided.
Theorem 2.2.There does not exist a polynomial affine translation surface with non-zero constant Gaussian curvature in E 3 .
Proof: Let M be a polynomial affine translation surface with constant Gaussian curvature.From (2.2) we have f ′′ g ′′ = 0 .Differentiating (2.2) with respect to y, we get Denoting f ′ and g ′ by α and β, respectively, we obtain ii.Suppose that n > m (≥ 2) Using similar way, this case cannot occur.
ii. Suppose that m = n (≥ 2) This case can be treated in similar way.ii.n > m = 1.From the similar way, this case cannot occur.Case 3. m ≥ n = 0(or n ≥ m = 0) this situation is not possible since f ′′ g ′′ = 0.
So, in every case, we obtain that there is no a polynomial affine translation surfaces with constant Gaussian curvature.✷ So, the following result can be given Corollary 2.3.If the Gaussian curvature of a polynomial affine translation surfaces in E 3 is equal to a constant, the constant must be zero.
Theorem 2.4.There does not exist a polynomial affine translation surface with constant mean curvature in E 3 .
Proof: Suppose that M is a polynomial affine translation surface with constant mean curvature.Differentiating equation (2.3) with respect to y, we get Denoting f ′ by α and g ′ by β we have Let us assume α and β are polynomials given by where b m and c n are non-zero constants.Substiuting α and β in (2.7) we get a polynomial expression in u and v vanishing identically that is all the coefficients are zero.Let us consider some cases of equation (2.7) : ii. Suppose that n > m (≥ 2) and m = n (≥ 2).It is easy to see that these cases can be treated using similar method mentioned above.
n .Then this case cannot occur since b, c n = 0 .
ii. m > n = 1.Using similar way, this case cannot occur.
Case 3. m, n ≥ 0 i. m ≥ n = 0. Then β is a constant, so the equation (2.7) is satisfied.But if β is constant from the equation (2.3) α is not be a polynomial.It is a contradiction, so this situation cannot occur.
ii. n ≥ m = 0. Then α (α = b ) is constant, so the equation (2.7) can rewrite with this case in the following way Using the same idea like in case 1,2 we can say that this situation cannot occur since b m = 0. So, the proof is completed.✷ Then the following result is given.It can be plotted as in Fig. 3.

A Further Application
As a further application we can choose the functions α and β as the exponential ones, i.e., α = c 1 e u and β = c 2 e v where c 1 and c 2 are real numbers and c 1 , c 2 = 0. Then the equation (2.6) can be written It is easy to see that the coefficients c 1 and c 2 have to be zero in order to satisfy the above equation, but this is not possible.Then we have the following:

. 6 ) 6 )
Polynomial Affine Translation Surfaces in Euclidean 3-Space 199 Suppose that the polynomials α and β are given byα = b m u m + b m−1 u m−1 + ... + b 1 u + b 0 and β = c n v n + c n−1 v n−1 + ... + c 1 v + c 0where b m and c n are non-zero constants.Replacing α and β in (2.6) we get a polynomial expression in u and v vanishing identically, i.e., all the coefficients are zero.Let us consider some cases of equation (2.Case 1. m, n ≥ 2 i.Suppose that m > n (≥ 2).The dominant term according to u 2m v n−2 which comes fromβ ′′ + β ′′ α 2 + 2aαββ ′′ having the coefficient b 2 m c n n (n − 1).This cannot vanish since b m , c n = 0 and m > n ≥ 2 .

Case 2 .
m, n ≥ 1 i. n > m = 1 .We get α = bu + d with real constants b, d and b = 0 .If we consider this situation in equation (2.7), the coefficient of highest degree

202H.Corollary 3 . 1 . 3 .
Gün Bozok and M. Ergüt There does not exist an exponential affine translation surface with non-zero constant Gaussian curvature in E If we get the functions α and β as the exponential ones again, the equation (2.7) can be written2c 1 c 2 2 e u e 2v + c 2 e v 1 + a 2 + c 2 1 e 2u 1 + (c 1 e u + ac 2 e v ) 2 + c 2 2 e 2v −3[ c 1 e u 1 + c 2 2 e 2v + c 2 e v 1 + a 2 + c 2 1 e 2u (c 1 e u + ac 2 e v ) ac 2 e v + c 2 2e 2v ] = 0 Considering the same technique mentioned before we obtain the following result: Corollary 3.2.There does not exist an exponential affine translation surface with non-zero constant mean curvature in E 3 .