  We characterize the triples (X,L,H), consisting of holomorphic line bundles L
and H on a complex projective manifold X, such that for some positive integer
k, the k-th holomorphic jet bundle of L, J_k(L), is isomorphic to a direct sum
H+...+H. Given the geometrical constrains imposed by a projectivized line
bundle being a product of the base and a projective space it is natural to
expect that this would happen only under very rare circumstances. It is shown,
in fact, that X is either an Abelian variety or projective space. In the former
case L\cong H is any line bundle of Chern class zero. In the later case for k a
positive integer, L=O_{P^n}(q) with J_k(L)=H+...+H if and only if
H=O_{P^n}(q-k) and either q\ge k or q\le -1.
