Homogeneous hypersurfaces of the four-dimensional Thurston geometry ${\rm Sol_0^4}$

TL;DR

This paper classifies hypersurfaces with constant principal curvatures in the four-dimensional Thurston geometry ${ m Sol_0^4}$ and provides a complete classification of homogeneous hypersurfaces, solving an open problem in the field.

Contribution

It offers the first classification of such hypersurfaces in ${ m Sol_0^4}$, advancing understanding of geometric structures in this Thurston geometry.

Findings

Complete classification of hypersurfaces with constant principal curvatures in ${ m Sol_0^4}$.

Classification of all homogeneous hypersurfaces in ${ m Sol_0^4}$.

Resolution of an open problem posed by Erjavec and Inoguchi.

Abstract

In this paper, we classify hypersurfaces with constant principal curvatures in the four-dimensional Thurston geometry ${\rm Sol_0^4}$ under certain geometric conditions. As an application of the classification result, we give a complete classification of homogeneous hypersurfaces in ${\rm Sol_0^4}$, which solves a problem raised by Erjavec and Inoguchi (Problem 6.4 of [J. Geom. Anal. 33, Art. 274, (2023)]).