Classification of homogeneous hypersurfaces in some noncompact symmetric spaces of rank two
Ivan Solonenko
TL;DR
The paper classifies homogeneous hypersurfaces in three noncompact rank-two symmetric spaces, namely SL(3,H)/Sp(3), SO(5,C)/SO(5), and Gr*(2,C^{n+4}). Building on the Berndt–Tamaru framework for cohomogeneity-one actions, it combines canonical extension from boundary components, reductive actions, and protohomogeneous subspace analysis to enumerate all action types, and it shows that the nilpotent construction does not yield new actions for these spaces. The results reinforce the conjecture that nilpotent construction mainly produces new examples in exceptional spaces and clarify the full action landscape for these rank-two examples. The work also includes an error-correction note and rank-one insights that support a uniform treatment of extending boundary actions to higher-rank noncompact symmetric spaces.
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 $\mathrm{Gr}^*(2,\mathbb{C}^{n+4}) = \mathrm{SU}(n+2,2)/\mathrm{S}(\mathrm{U}(n+2)\mathrm{U}(2)), \, n \geqslant 1$.