  In this work we deal with vector bundles of rank two on a Fano manifold $X$
with $b_2=b_4=1$. We study the nef and pseudoeffective cones of the
corresponding projectivizations and how these cones are related to the
decomposability of the vector bundle. As consequences, we obtain the complete
list of $\mathbb{P}^1$-bundles over $X$ that have a second
$\mathbb{P}^1$-bundle structure, classify all the uniform rank two vector
bundles on this class of Fano manifolds and show the stability of
indecomposable Fano bundles (with one exception on $\mathbb{P}^2$).
