Let $M$ be a smooth manifold with tangent bundle $TM$ and cotangent bundle $TM^*$ and $\psi\in TM^*$ a one-form. We denote the quotient manifold of $M$ by the free and proper $G$-action $\varphi$ as $\bar M=M/G$. The projection map $\pi:M\mapsto \bar M$ is a smooth submersion. I would like to know how to project the one-form $\psi$ on $\bar M$ i.e. define a one-form $\bar \psi\in T\bar M^*$ by projecting $\psi$ with $\pi$ and eventually to know the conditions $\psi$ has to satisfy in order for $\bar \psi$ to be well-defined. In the case of a vector field, the pushforward furnishes a natural way in order to define vectors at each point of $\bar M$ but as the pullback operates backwards, it doesn't seem suited to define one-forms from $M$ to $\bar M$. Does anyone has any suggestions or references ? Thank you very much.
Edit: More precisely, I'm interested in a manifold $M$ quotiented by the action defined by the flow of a given vector field so the group $G=(R,+)$ which is not discrete and the projection map $\pi:M\mapsto \bar M=M/G$ is not an immersion. I dispose of a one-form or covector field on $M$, $\psi:TM\mapsto C^\infty$ and would like to know the conditions on $\psi$ in order for $\psi$ to induce a well-defined one-form field $\bar \psi:T\bar M\mapsto C^\infty$ on the quotient manifold $\bar M$.