Translation surfaces in Euclidean space with constant Gaussian curvature

TL;DR

This paper proves that the only translation surfaces in three-dimensional Euclidean space with constant Gaussian curvature are cylindrical surfaces, which have zero Gaussian curvature, thus classifying such surfaces completely.

Contribution

It establishes a complete classification of translation surfaces with constant Gaussian curvature in Euclidean space, showing they must be cylindrical surfaces.

Findings

Only cylindrical surfaces have constant Gaussian curvature among translation surfaces.

Such surfaces necessarily have zero Gaussian curvature.

The classification is complete for translation surfaces in Euclidean space.

Abstract

We prove that the only surfaces in $3$-dimensional Euclidean space $\R^3$ with constant Gaussian curvature $K$ and constructed by the sum of two space curves are cylindrical surfaces, in particular, $K=0$.