  Here we consider the product of varieties with $n$-blocks collections . We
give some cohomological splitting conditions for rank 2 bundles. A
cohomological characterization for vector bundles is also provided. The tools
are Beilinson's type spectral sequences generalized by Costa and Mir\'o-Roig.
Moreover we introduce a notion of Castelnuovo-Mumford regularity on a product
of finitely many projective spaces and smooth quadric hypersurfaces in order to
prove two splitting criteria for vector bundle with arbitrary rank.
