Summary: A main objective in submanifold geometry is the classification of homogeneous hypersurfaces. Homogeneous hypersurfaces arise as principal orbits of cohomogeneity one actions, and so their classification is equivalent to the classification of cohomogeneity one actions up to orbit equivalence. Actually, the classification of cohomogeneity one actions in irreducible simply connected Riemannian symmetric spaces of rank 2 of noncompact type was obtained by J. Berndt and the second author [Int. J. Math. 23, No. 10, Paper No. 1250103, 35 p. (2012; Zbl 1262.53046)] (for complex hyperbolic two-plane Grassmannian \(\mathrm{SU}_{2,m}/\mathrm{S}(\mathrm{U}_{2}{\cdot} \mathrm{U}_{m}\)). From this classification, in [Adv. Appl. Math. 50, No. 4, 645–659 (2013; Zbl 1279.53051)] the second author classified real hypersurfaces with isometric Reeb flow in \(\mathrm{SU}_{2,m}/\mathrm{S}(\mathrm{U}_{2}{\cdot} \mathrm{U}_{m})\), \(m \geq 2\). Each one can be described as a tube over a totally geodesic \(\mathrm{SU}_{2,m-1}/\mathrm{S}(\mathrm{U}_{2}{\cdot} \mathrm{U}_{m-1})\) in \(\mathrm{SU}_{2,m}/\mathrm{S}(\mathrm{U}_{2}{\cdot} \mathrm{U}_{m})\) or a horosphere whose center at infinity is singular. By using this result, we want to give another characterization for these model spaces by the Reeb invariant shape operator, that is, \(\mathcal{L}_{{\xi} }A = 0\).
For the entire collection see [Zbl 1304.53004].