REEB LIE DERIVATIVES ON REAL HYPERSURFACES IN COMPLEX HYPERBOLIC TWO-PLANE GRASSMANNIANS

Abstract

In complex two plane Grassmannians, it is known that a real hypersurface satisfying the condition $(\hat \L ^{(k)}_{\xi}R_{\xi})Y=(\L_{\xi}R_{\xi})Y$ is locally congruent to an open part of a tube around a totally geodesic $G_2({\mathbb C}^{m+1})$ in $G_2({\mathbb C}^{m+2})$. In this paper, as an abient space, we consider a complex hyperbolic two-plane Grassmannian $SU_{2,m}/S(U_{2}{\cdot}U_{m})$ and give a complete classification of a real hypersurface in $SU_{2,m}/S(U_{2}{\cdot}U_{m})$ with the above condition.