Einstein–Weyl structures on real hypersurfaces of complex two-plane Grassmannians

Abstract

. We study real Hopf hypersurfaces with Einstein–Weyl structures in the complex two-plane Grassmannian G 2 ( C m +2 ) , m ≥ 3 . First we prove that a real Hopf hypersurface with a closed Einstein–Weyl structure W = ( g, θ ) is of type (B) if ∇ ξ θ = 0 , where ξ 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, kη ) , where k is a non-zero constant and η is a one-form dual to ξ . Finally, it is proved that a real Hopf hypersurface with two closed Einstein–Weyl structures W ± = ( g, ± θ ) is of type (A) or type (B). ,