Isometric Reeb flow in complex hyperbolic quadrics

Abstract

We classify real hypersurfaces with isometric Reeb flow in the complex hyperbolic quadrics ${Q^{\ast }}^m=S{O}_{2,m}^o/S{O}_{m}S{O}_2$, $m\ge 3$. We show that $m$ is even, say $m=2k$, and any such hypersurface becomes an open part of a tube around a $k$-dimensional complex hyperbolic space $\mathbb{C}{H}^k$ which is embedded canonically in ${Q^{\ast }}^{2k}$ as a totally geodesic complex submanifold or a horosphere whose center at infinity is $\mathfrak{A}$-isotropic singular. As a consequence of the result, we get the non-existence of real hypersurfaces with isometric Reeb flow in odd-dimensional complex quadrics ${Q^{\ast }}^{2k+1}$, $k\ge 1$.

arXiv admin note: substantial text overlap with arXiv:1301.0411