Homogeneous hypersurfaces of the four-dimensional Thurston geometries $\mathrm{Sol}_1^4$, $\mathrm{Sol}_{m,n}^4$ and $\mathrm{Nil}^4$

Abstract

In this paper, we focus on the four-dimensional Thurston geometries whose isometry groups are four-dimensional, namely $\mathrm{Sol}_1^4$, $\mathrm{Sol}_{m,n}^4$ and $\mathrm{Nil}^4$. We classify homogeneous hypersurfaces in the above three manifolds.