Real Hypersurfaces in the Complex Hyperbolic Quadric with Commuting Ricci Tensor

Abstract

We study Ricci-commuting real hypersurfaces in the complex hyperbolic quadric $${{Q^m}^*} = SO^{o}_{2,m}/SO_mSO_2$$ , $$m \ge 3$$ . The commuting Ricci tensor shows that the unit normal vector field N of $${Q^m}^*$$ is singular, that is, $$AN=N$$ or $$N=\frac{1}{\sqrt{2}}(Z_1+JZ_2)$$ , $$Z_1,Z_2 \in V(A)$$ for a complex conjugation $$A \in {\mathfrak {A}}$$ , where $$\mathfrak {A}$$ denotes the set of all complex conjugations in $${Q^m}^*$$ . Then according to each case, we give a classification of real hypersurfaces having a commuting Ricci tensor in $${Q^m}^*$$ .