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

Abstract

A real hypersurface $M$ in the complex quadric $Q^{m}=SO_{m+2}/SO_mSO_2$ 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 $M$, $\hat{\nabla}^{(k)}$. If $\nabla$ denotes the Levi-Civita connection on $M$, we introduce the concepts of $(\hat{\nabla}^{(k)},\nabla)$-Codazzi and $(\hat{\nabla}^{(k)},\nabla)$-Killing shape operator $S$ of the real hypersurface and classify real hypersurfaces in $Q$ satisfying any of these conditions.