On translation surfaces in 4-dimensional Euclidean space

Abstract

We consider translation surfaces in Euclidean spaces. Firstly, we give some results of translation surfaces in the 3-dimensional Euclidean space E³. Further, we consider translation surfaces in the 4-dimensional Euclidean space E⁴. We prove that a translation surface is flat in E⁴ if and only if it is either a hyperplane or a hypercylinder. Finally we give necessary and sufficient condition for a quadratic triangular Bézier surface in E⁴ to become a translation surface.