Some Real Hypersurfaces in Complex and Complex Hyperbolic Quadrics

Abstract

On a real hypersurface in the complex quadric or the complex hyperbolic quadric we can consider the Levi-Civita connection and, for any nonnull real number $k$, the $k$-th generalized Tanaka-Webster connection. We also have a differential operator of first order of Lie type associated to the $k$-th generalized Tanaka-Webster 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 they act on 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.