Classification of separable hypersurfaces with constant sectional curvature

TL;DR

This paper fully classifies separable hypersurfaces with constant sectional curvature in Euclidean spaces, identifying specific families for null curvature and proving hyperspheres are unique for nonzero curvature.

Contribution

It extends the classification of such hypersurfaces to higher dimensions and characterizes hyperspheres as the only nonzero constant curvature separable hypersurfaces.

Findings

Null sectional curvature hypersurfaces form three specific families.

Hyperspheres are the only separable hypersurfaces with nonzero constant sectional curvature.

Complete classification in all dimensions n ≥ 3.

Abstract

In this paper, we give a full classification of the separable hypersurfaces of constant sectional curvature in the Euclidean $n$-space $\mathbb{R}^n$. In dimension $n=3$, this classification was solved by Hasanis and L\'opez [Manuscripta Math. 166, 403-417 (2021)]. When $n>3$, we prove that the separable hypersurfaces of null sectional curvature are three particular families of such hypersurfaces. Finally, we prove that hyperspheres are the only separable hypersurfaces with nonzero constant sectional curvature.