New characterizations of ruled real hypersurfaces in 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 constant k and any symmetric tensor field of type (1, 1) L on M, we can define two tensor fields of type (1, 2) on M, L-F((k)) and (LT)-T-(k) , related to both connections. We study the behaviour of the structure operator f with respect to such tensor fields in the particular case of L = A, the shape operator of M, and obtain some new characterizations of ruled real hypersurfaces in complex projective space.