Morse-Bott Volume Forms

A Morse-Bott volume form on a manifold is a top-degree form which vanishes along a non-degenerate critical submanifold. We prove that two such forms are diffeomorphic (by a diffeomorphism fixed on the submanifold) provided that their relative cohomology classes with respect to the submanifold coincide. For a zero submanifold of codimension at least 2, this means that two Morse-Bott volume forms with the same zero set are diffeomorphic if and only if they have equal total volumes. We show how "Moser's trick" for establishing equivalence of non-degenerate volume forms can be adapted to this setting.