Real hypersurfaces in complex hyperbolic two-plane Grassmannians with parallel Ricci tensor (Q2929416)

Let \(\mathrm{SU}_{2,m}/\mathrm{S}(\mathrm{U}_2 \cdot \mathrm{U}_m)\) denote the set of all complex \(2\)-dimensional linear subspaces in the indefinite complex Euclidean spaces \(\mathbb{C}_2^{m+2}\), called the \textit{complex hyperbolic two-plane Grassmannian}. The authors characterize a connected hypersurface in \(\mathrm{SU}_{2,m}/\mathrm{S}(\mathrm{U}_2 \cdot \mathrm{U}_m)\) with \(m \geq 2\), which has the property that, the maximal complex subbundle \(\mathfrak{C}\) of \(TM\) and the maximal quaternionic subbundle \(\mathfrak{Q}\) of \(TM\) are both invariant under the shape operator of \(M\). This classification leads the authors to prove the following: There does not exist any Hopf real hypersurface in complex hyperbolic two-plane Grassmannian \(\mathrm{SU}_{2,m}/\mathrm{S}(\mathrm{U}_2 \cdot \mathrm{U}_m)\), \(m \geq 3\), with parallel Ricci curvature.