Real hypersurfaces with isometric Reeb flow in complex quadrics (Q2853384)

In 1975, Masafumi Okumura obtained a full classification of real hypersurfaces with isometric Reeb flow in the complex projective space \(\mathbb CP^m\). For the complex 2-plane Grassmannian \(G_2(\mathbb C^{m+2})=\mathrm{SU}_{m+2}/\mathrm{S}(\mathrm{U}_m\mathrm{U}_2)\) the classification was obtained by the authors in [Monatsh. Math. 137, No. 2, 87--98 (2002; Zbl 1015.53034)]. In this paper, the authors classify real hypersurfaces with isometric Reeb flow in the complex quadrics \(Q^m=\mathrm{SO}_{m+2}/\mathrm{SO}_m\mathrm{SO}_2\), \(m\geq 3\). They show that \(m\) is even, say \(m=2k\), and any such hypersurface is an open part of a tube around a \(k\)-dimensional complex projective space \(\mathbb CP^k\) which is embedded canonically in \(Q^{2k}\) as a totally geodesic complex manifold. As a consequence, the authors get the non-existence of real hypersurfaces with isometric Reeb flow in odd-dimensional complex quadrics \(Q^{2k+1}\), \(k\geq 1\). It is remarkable that the odd-dimensional complex quadrics are the first examples of homogeneous Kähler manifolds which do not admit a real hypersurface with isometric Reeb flow.