Affine translation surfaces in the isotropic 3-space

TL;DR

This paper classifies and describes affine translation surfaces in isotropic 3-space, focusing on Weingarten conditions and specific equations involving the position vector and Laplace operator.

Contribution

It provides a comprehensive classification of affine translation surfaces in isotropic 3-space satisfying Weingarten conditions and certain differential equations.

Findings

Classification of Weingarten affine translation surfaces

Explicit descriptions of surfaces satisfying Laplace-related equations

New insights into the geometry of isotropic 3-space surfaces

Abstract

The isotropic 3-space \mathbb{I}^{3} is a real affine 3-space endowed with the metric dx^{2}+dy^{2}. In this paper we describe Weingarten and linear Weingarten affine translation surfaces in \mathbb{I}^{3}. Further we classify the affine translation surfaces in \mathbb{I}^{3} that satisfy certain equations in terms of the position vector and the Laplace operator.