Suppose $M$ is a compact smooth manifold, and let $R$ be a nowhere-vanishing vector field. Then in the irregular scenario, the closure of $R$-orbits is a $r$-torus $T^r$.
Now suppose there is a function $f$ which is invariant under $R$. Then by continuity, $f$ is actually constant on each $T^k$ orbit (correct me if I'm wrong).
I wonder if similar holds if $g$ is a tensor (or differential form) which is invariant under $R$, namely if it is invariant under $T^r$ action?