Flag bundles on Fano manifolds

As an application of a recent characterization of complete flag manifolds as Fano manifolds having only P1-bundles as elementary contractions, we consider here the case of a Fano manifold X of Picard number one supporting an unsplit family of rational curves whose subfamilies parametrizing curves through a fixed point are rational homogeneous, and we prove that X is homogeneous. In order to do this, we first study minimal sections on flag bundles over the projective line, and discuss how Grothendieck's theorem on principal bundles allows us to describe a flag bundle upon some special sections.