Publishing house / Journals and Serials / Colloquium Mathematicum / All issues

Abstract

We study real Hopf hypersurfaces with Einstein–Weyl structures in the complex two-plane Grassmannian $G_2(\mathbb {C}^{m+2})$, $m\geq 3$. First we prove that a real Hopf hypersurface with a closed Einstein–Weyl structure $W=(g,\theta )$ is of type (B) if $\nabla _\xi \theta =0$, where $\xi $ denotes the Reeb vector field of the hypersurface. Next, for a Hopf hypersurface with non-vanishing geodesic Reeb flow, we prove that there does not exist an Einstein–Weyl structure $W=(g,k\eta )$, where $k$ is a non-zero constant and $\eta $ is a one-form dual to $\xi $. Finally, it is proved that a real Hopf hypersurface with two closed Einstein–Weyl structures $W^\pm =(g,\pm \theta )$ is of type (A) or type (B).