On projective nanifolds with two P-bundle structures

We investigate whether a projective manifold V which is a P-bundle over a projective manifold X with the same homology type of P^r can admit another P-bundle structure over some projective manifold Y; morover, in the affirmative case we find restrictions on the topology of Y. Among the corollaries we prove that if V is a P-bundle over P^4 and over a 4-fold Y not of general type, then either V=P^4 \times P^4, V = P(T_{P^4}), or, possibly, Y is a rational Fano 4-fold many properties of which are known. Further generalizations naturally arising from the geometry of flag manifolds are discussed.