Classification of homogeneous hypersurfaces in some noncompact symmetric spaces of rank two

TL;DR

This paper classifies homogeneous hypersurfaces in specific noncompact symmetric spaces of rank two, providing a comprehensive understanding of their geometric structures up to isometric congruence.

Contribution

It offers the first complete classification of homogeneous hypersurfaces in these particular noncompact symmetric spaces of rank two.

Findings

Classification of hypersurfaces in $ ext{SL}(3, ext{H})/ ext{Sp}(3)$

Classification in $ ext{SO}(5, ext{C})/ ext{SO}(5)$

Classification in $ ext{Gr}^*(2, ext{C}^{n+4})$ for $n \u003e= 1$

Abstract

We classify, up to isometric congruence, the homogeneous hypersurfaces in the Riemannian symmetric spaces $\mathrm{SL}(3,\mathbb{H})/\mathrm{Sp}(3), \hspace{1pt} \mathrm{SO}(5,\mathbb{C})/\mathrm{SO}(5),$ and $\mathrm{Gr}^*(2,\mathbb{C}^{n+4}) = \mathrm{SU}(n+2,2)/\mathrm{S}(\mathrm{U}(n+2)\mathrm{U}(2)), \, n \geqslant 1$.