Affine translation hypersurfaces in Euclidean and isotropic spaces

TL;DR

This paper generalizes affine translation hypersurfaces to higher dimensions in Euclidean and isotropic spaces, characterizing those with constant Gauss-Kronocker curvature and exploring their properties.

Contribution

It extends the concept of affine translation surfaces to higher dimensions and analyzes their curvature properties in Euclidean and isotropic spaces.

Findings

Affine translation hypersurfaces with constant curvature are cylinders.

Such hypersurfaces in isotropic spaces satisfy specific curvature and Laplacian conditions.

The paper provides a classification of these hypersurfaces based on curvature.

Abstract

In this paper, we extend the notion of affine translation surfaces introduced by Liu and Yu (Proc. Japan Acad. Ser. A Math. Sci. 89, 111--113, 2013) in a Euclidean space R^{3} to higher dimensional ambient spaces. We provide that an affine translation hypersurface of constant Gauss-Kronocker curvature K_{0} in R^{n+1} is a cylinder, i.e. K_{0}=0. As further applications we describe such hypersurfaces in the isotropic spaces satisfying certain conditions on the isotropic curvatures and the Laplacian.