Parameter dependent pull-back of closed differential forms and invariant integrals

Abstract

We prove, given a closed differential $k$-form $\omega$ in an arbitrary open set $D \subset {\mathbb R}^n$, and a parameter dependent smooth map $F(\cdot,\lambda)$ from an arbitrary open set $G \subset {\mathbb R}^m$ into $D$, that the derivative with respect to $\lambda$ of the pull-back $F(\cdot,\lambda)^{*}\omega$ is exact in $G$. We give applications to various theorems in topology, dynamics and hydrodynamics.