We investigate translation hypersurfaces in the $(n+1)$-dimensional Euclidean space whose Gauss-Kronecker curvature depends on either its first p or on its second q variables. These hypersurfaces are the graph of the sum of two functions in p and q independent variables respectively and with $n=p+q$. We prove that under this condition, the Gauss-Kronecker curvature is constant zero. On other side, if the mean curvature is nowhere zero and it depends on either its first p or on its second q variables, we get again that the Gauss-Kronecker curvature is constant zero.

中文翻译：

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平移超曲面，其曲率部分取决于其变量

我们调查翻译中的超曲面 $（ñ+\mathrm{1个}）$维高斯-克罗内克曲率取决于其第一个p或第二个q变量的二维欧几里德空间。这些超曲面分别是p和q自变量中两个函数之和的图形$ñ=p+q$。我们证明在这种情况下，高斯-克罗内克曲率恒定为零。另一方面，如果平均曲率在任何地方都不为零，并且取决于它的第一个p或第二个q变量，我们将再次得到高斯-克罗内克曲率是恒定的零。