Some Real Hypersurfaces in Complex and Complex Hyperbolic Quadrics

We will consider real hypersurfaces in the complex quadric or the complex hyperbolic quadric. From the almost contact metric structure of such hypersurfaces, we can define on any real hypersurface, for any nonnull real number k, a kth generalized Tanaka–Webster connection and a differential operator of first order of Lie type associated with such a connection. We classify real hypersurfaces in the complex quadric and the complex hyperbolic quadric for which the Lie derivative and the Lie-type differential operator coincide when we apply them to the shape operator of the real hypersurface either in the direction of the structure vector field or in any direction of the maximal holomorphic distribution.