Some Characterizations of Real Hypersurfaces in Complex Hyperbolic Two-Plane Grassmannians

Abstract

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 Y.J. Suh (for complex hyperbolic two-plane Grassmannian $SU_{2,m}/S(U_{2}\cdot U_{m}$), (Berndt and Suh, Int. J. Math. 23, 1250103 (35pages), 2012)). From this classification, in (Suh, Adv. Appl. Math. 50, 645–659, 2013) Suh classified real hypersurfaces with isometric Reeb flow in $SU_{2,m}/S(U_{2}\cdot U_{m})$, m ≥ 2. Each one can be described as a tube over a totally geodesic $SU_{2,m-1}/S(U_{2}\cdot U_{m-1})$ in $SU_{2,m}/S(U_{2}\cdot U_{m})$ 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, $\mathcal{L}_{\xi }A = 0$.