On the Geometry of the Second Fundamental Form of Translation Surfaces in E3

TL;DR

This paper investigates the geometric properties of translation surfaces in Euclidean 3-space, focusing on the second fundamental form, classifying certain types, and establishing non-existence results for specific curvature conditions.

Contribution

It provides a classification of translation surfaces with proportional second fundamental form and proves the non-existence of II-minimal translation surfaces in E3.

Findings

Polynomial translation surfaces with vanishing second Gaussian curvature do not exist.

Certain translation surfaces with proportional KII and H are classified.

No II-minimal translation surfaces exist in Euclidean 3-space.

Abstract

In this paper we study the second fundamental form of translation surfaces in E3. We give a non-existence result for polynomial translation surfaces in E3 with vanishing second Gaussian curvature KII. We classify those translation surfaces for which KII and H are proportional. Finally we obtain that there are no II-minimal translation surfaces in the Euclidean 3-space.