Lie 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. Associated with the generalized Tanaka-Webster connection we can define a differential operator of first order. For any nonnull real number k and any symmetric tensor field of type (1,1) B on M, we can define a tensor field of type (1,2) on M, B-T((k)), related to Lie derivative and such a differential operator. We study symmetry and skew symmetry of the tensor A(T)((k)) associated with the shape operator A of M.