Quaternionic contact hypersurfaces in hyper-Kähler manifolds

Abstract

We describe explicitly all quaternionic contact hypersurfaces (qc-hypersurfaces) in the flat quaternion space $$\mathbb {H}^{n+1}$$ and the quaternion projective space. We show that up to a quaternionic affine transformation a qc-hypersurface in $$\mathbb {H}^{n+1}$$ is contained in one of the three qc-hyperquadrics in $$\mathbb {H}^{n+1}$$ . Moreover, we show that an embedded qc-hypersurface in a hyper-Kahler manifold is qc-conformal to a qc-Einstein space and the Riemannian curvature tensor of the ambient hyper-Kahler metric is degenerate along the hypersurface.