Recent Results on Real Hypersurfaces in Complex Quadrics

Abstract

In this survey article, first we introduce the classification of homogeneous hypersurfaces in some Hermitian symmetric spaces of rank 2. Second, by using the isometric Reeb flow, we give a complete classification for hypersurfaces M in complex two-plane Grassmannians \(G_2({\mathbb C}^{m+2})=SU_{2+m}/S(U_{2}U_{m})\), complex hyperbolic two-plane Grassmannians \(G_{2}^{*}({\mathbb C}^{m+2})=SU_{2,m}/S(U_{2}U_{m})\), complex quadric \(Q^m={ SO}_{m+2}/SO_{m}SO_{2}\) and its dual \(Q^{m *}= SO_{m,2}^{o}/SO_{m}SO_{2}\). As a third, we introduce the classifications of contact hypersurfaces with constant mean curvature in the complex quadric \(Q^m\) and its noncompact dual \(Q^{m *}\) for \(m \ge 3\). Finally we want to mention some classifications of real hypersurfaces in the complex quadrics \(Q^m\) with Ricci parallel, harmonic curvature, parallel normal Jacobi, pseudo-Einstein, pseudo-anti commuting Ricci tensor and Ricci soliton etc.