Translation hypersurfaces with constant curvature in 4-dimensional isotropic space

TL;DR

This paper classifies and analyzes translation hypersurfaces with constant curvature in 4-dimensional isotropic space, extending previous work to three additional types beyond the known four, and explores their geometric properties.

Contribution

It investigates three new types of translation hypersurfaces with constant curvature in 4D isotropic space, expanding the classification beyond previously studied cases.

Findings

Four non-equivalent types of translation hypersurfaces identified

Explicit conditions for constant curvature hypersurfaces derived

Extension of previous results to additional hypersurface types

Abstract

There exist four non-equivalent types of the translation hypersurfaces in the 4-dimensional isotropic space $\mathbb{I}^{4}$ generated by translating the curves lying in perpendicular $k-$planes $\left(k=2,3\right)$, due to its absolute figure. In arbitrary dimensional case; constant Gauss-Kronecker and mean curvature translation hypersurfaces of type 1, i.e. the hypersurfaces whose the translating curves lie in perpendicular isotropic $2- $planes, were investigated by the same authors in \cite{AO}. The present study concerns such hypersurfaces in $\mathbb{I}^{4}$ of other three types.