New results on derivatives of the shape operator of a real hypersurface in a complex projective space

Abstract

We consider real hypersurfaces $M$ in complex projective space equipped with both the Levi-Civita and generalized Tanaka-Webster connections. For any nonnull real number $k$ and any symmetric tensor field of type (1,1) $L$ on $M$ we can define a tensor field of type (1,2) on $M$, $L^{(k)}_F$, related to both connections. We study symmetry and skewsymmetry of the tensor $A^{(k)}_F$ associated to the shape operator$A$ of $M$.