  As an application of a recent characterization of complete flag manifolds as
Fano manifolds having only ${\mathbb P}^1$-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.
