Affine Translation Surfaces with Constant Gaussian Curvature

Abstract

Abstract. We study a–ne translation surfaces in R 3 and get a complete classiflcation ofsuch surfaces with constant Gauss-Kronecker curvature. 1. IntroductionA surface in E 3 is called a translation surface if it is obtained as a graph of a func-tion F ( x;y ) = p ( x )+ q ( y ), where p ( x ) and q ( y ) are difierentiable functions. It’s wellknown that a minimal translation surface in the Euclidean space E 3 must be a planeor a Scherk surface, which is the graph of the function F ( x;y ) = ln(cos x= cos y ),the only doubly periodic minimal translation surface.In this note, we study nondegenerate translation surfaces in a–ne space R 3 .This class of surfaces has been studied previously by many geometers. F. Manhart[3] classifled all the nondegenerate a–ne minimal translation surfaces in a–ne spaceR 3 . Further treatments are due to H. F. Sun [5], who classifled the nondegeneratea–ne translation surface with nonzero constant mean curvature in R 3 . Later on,Sun and Chen extended this into the case of hypersurfaces [6]. On the other hand,Binder [1] classifled locally symmetric a–ne translation surfaces in R