New Results on Derivatives of the Shape Operator of Real Hypersurfaces in the Complex Quadric

A real hypersurface M in the complex quadric Q^m=SO₌+₂/SOₘSO₂ inherits an almost contact metric structure. This structure allows to define, for any nonnull real number k, the so called k-th generalized Tanaka-Webster connection on M, ^ (k). If denotes the Levi-Civita connection on M, we introduce the concepts of (^ (k), ) -Codazzi and (^ (k), ) -Killing shape operator S of the real hypersurface and classify real hypersurfaces in Q satisfying any of these conditions.