The almost contact metric structure that we have on a real hypersurface $M$ in the complex quadric $Q^{ m}=SO_{ m+2}/SO_mSO_2$ allows us to define, for any nonnull real number $k$, the $k$-th generalized Tanaka-Webster connection on $M$, $\hat{ \nabla}^{ (k)}$. Associated to this connection we have Cho and torsion operators, $F_X^{ (k)}$ and $T_X^{ (k)}$, respectively, for any vector field $X$ tangent to $M$. From them and for any symmetric operator $B$ on $M$ we can consider two tensor fields of type (1,2) on $M$ that we will denote by $B_F^{ (k)}$ and $B_T^{ (k)}$, respectively. We will classify real hypersurfaces $M$ in $Q^m$ for which any of those tensors identically vanishes, in the particular case of $B$ being the structure Lie operator $L_{ \xi}$ on $M$.
Categories: math.DG