  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 $L_T^{(k)}$,
related to both connections. We study the behaviour of the structure operator
$\phi$ 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.
