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

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