Curvature on reductive homogeneous spaces

In this work, we consider the general flag manifold $\vb_\Theta$ as a naturally reductive homogeneous space endows with an$U$--invariant metric $\Lambda^{\Theta}$ and an invariant almost-complex structure $J^\Theta$. Our central reference is the section 2 in chapter X of \cite{kn}. The main objective of this work is to explore the form for the {\em Riemannian connection} associated with the metric $\Lambda^{\Theta}$ in order to calculate some classes of curvatures which allow to compare the results that appear in the geometric flag manifolds with characterizations already existing (see, for example, \cite{siu}, \cite{eell2}).