Classification results on surfaces in the isotropic 3-space

TL;DR

This paper classifies surfaces in isotropic 3-space based on constant relative curvature and isotropic mean curvature, including helicoidal surfaces, providing new insights into their geometric properties.

Contribution

It introduces classifications of surfaces in isotropic 3-space with constant curvature, focusing on helicoidal surfaces and special curves, which is a novel geometric analysis.

Findings

Classified surfaces with constant relative curvature in I^3

Classified surfaces with constant isotropic mean curvature in I^3

Analyzed special curves on helicoidal surfaces in I^3

Abstract

The isotropic 3-space I^3 which is one of the Cayley--Klein spaces is obtained from the Euclidean space by substituting the usual Euclidean distance with the isotropic distance. In the present paper, we give several classifications on the surfaces in I^3 with the constant relative curvature (analogue of the Gaussian curvature) and the constant isotropic mean curvature. In particular, we classify the helicoidal surfaces in I^3 with constant curvature and analyze some special curves on these.