On Rank two vector bundles on Fano manifolds

In this work we deal with vector bundles of rank two on a Fano manifold X with H^2(X; Z) \simeq Z, H^4(X; Z) \simeq Z. 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 P1-bundles over X that have a second P1-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 P2).