Isometric Reeb flow in complex hyperbolic quadrics

TL;DR

This paper classifies real hypersurfaces with isometric Reeb flow in complex hyperbolic quadrics, showing they are tubes around certain complex hyperbolic spaces or horospheres, and proves such hypersurfaces do not exist in odd-dimensional cases.

Contribution

It provides a complete classification of real hypersurfaces with isometric Reeb flow in complex hyperbolic quadrics, identifying their geometric structure and non-existence in odd dimensions.

Findings

Hypersurfaces occur only in even dimensions, with specific geometric structures.

Such hypersurfaces are tubes around complex hyperbolic spaces or horospheres.

No such hypersurfaces exist in odd-dimensional complex quadrics.

Abstract

We classify real hypersurfaces with isometric Reeb flow in the complex hyperbolic quadrics ${Q^*}^{m} = SO^{o}_{2,m}/SO_mSO_2$, $m \geq 3$. 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 ${Q^*}^{2k}$ as a totally geodesic complex submanifold or a horosphere whose center at infinity is $\frak A$-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 ${Q^*}^{2k+1}$, $k \geq 1$.