  We investigate manifolds obtained as a quotient of a doubly warped product.
We show that they are always covered by the product of two suitable leaves.
This allows us to prove, under regularity hypothesis, that these manifolds are
a doubly warped product up to a zero measure subset formed by an union of
leaves. We also obtain a necessary and sufficient condition which ensures the
decomposition of the whole manifold and use it to give sufficient conditions of
geometrical nature. Finally, we study the uniqueness of direct product
decomposition in the non simply connected case.
