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

TL;DR

This paper studies the geometric structure of real hypersurfaces in complex quadrics, introducing generalized connections and operators, and classifies hypersurfaces where certain tensor fields vanish, focusing on the structure Lie operator.

Contribution

It defines generalized Tanaka-Webster connections and associated tensors on hypersurfaces in complex quadrics, and classifies hypersurfaces with vanishing structure Lie operator tensors.

Findings

Classification of hypersurfaces with vanishing tensor fields

Introduction of generalized Tanaka-Webster connections on hypersurfaces

Analysis of structure Lie operator in the context of complex quadrics

Abstract

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 $M$.