On the structure Lie operator of a real hypersurface in the complex quadric

Abstract

The almost contact metric structure that we have on a real hypersurface $M$ in the complex quadric $Q^m=S{O}_{m+2}/S{O}_{m}S{O}_2$ 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 $M$.

9 pages