Classifications of translation surfaces in isotropic geometry with constant curvature

TL;DR

This paper classifies translation surfaces in isotropic geometry with constant curvature, focusing on cases where at least one translating curve is planar, expanding understanding of their geometric properties.

Contribution

It provides a comprehensive classification of translation surfaces with constant isotropic curvature when one translating curve is planar, a novel extension in isotropic geometry.

Findings

Classification of translation surfaces with constant isotropic Gaussian curvature

Classification of translation surfaces with constant isotropic mean curvature

Identification of specific geometric configurations under planar translating curve condition

Abstract

We classify translation surfaces in isotropic geometry with arbitrary constant isotropic Gaussian and mean curvature under the condition that at least one of translating curves lies in a plane.