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}_pεB, (τ_p)_pεB which guarantee the existence of a smooth family of diffeomorphisms {φ_p}_pεB such that _pω_p = τ_p for all p ε 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.