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

A real hypersurface M in the complex quadric Qᵐ=SOₘ₊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.