Rotational hypersurfaces with constant Gauss-Kronecker curvature

TL;DR

This paper investigates rotational hypersurfaces with constant Gauss-Kronecker curvature, solving related differential equations and discovering new classes of such hypersurfaces with specific geometric properties.

Contribution

It provides explicit solutions for generating curves of these hypersurfaces and introduces a novel class of non-compact hypersurfaces with finite volume and negative curvature.

Findings

Discovered a class of non-compact hypersurfaces with finite volume and negative curvature

Solved the ODEs governing the generating curves of these hypersurfaces

Analyzed geometric properties and extended to hypersurfaces with prescribed curvature

Abstract

We study rotational hypersurfaces with constant Gauss-Kronecker curvature. We solve the ODE for the generating curves of such hypersurfaces and analyze several geometric properties of such hypersurfaces. In particular, we discover a class of non-compact rotational hypersurfaces with constant and negative Gauss-Kronecker curvature and finite volume, which can be seen as the higher-dimensional generalization of the pseudo-sphere. Finally we investigate other types of rotational hypersurfaces with similar curvature constraints, including those with prescribed Gauss-Kronecker curvature.