Hopf hypersurfaces in complex hyperbolic space and submanifolds in indefinite complex 2-plane Grassmannian I

We define Gauss map from a real hypersurface in complex hyperbolic space to indefinite complex 2-plane Grassmannian. We show that if a real hypersurface is Hopf, then the image of the Gauss map is a half-dimensional regular submanifold and has a nice behavior under para-quaternionic Kähler structures of the Grassmannian. In particular if absolute value of the Hopf curvature of the Hopf hypersurface is greater (resp. smaller) than 2, then the Gauss image is totally complex (resp. totally para-complex) submanifold with respect to the para-quaternionic Kähler structure of indefinite complex 2-plane Grassmannian, provided that the induced metric is nondegenerate.