Moser–Greene–Shiohama stability for families

Let M be a noncompact oriented connected manifold and let B be a compact manifold. We give conditions on two smooth families of volume forms { ω p } p ∈ B , { τ p } p ∈ B which guarantee the existence of a smooth family of diffeomorphisms { ϕ p } p ∈ B such that ϕ ∗ p ω p = τ p for all p ∈ B . If B is a point, our result recovers a theorem of Greene and Shiohama from 1979, which itself extended a theorem of Moser for compact manifolds.