in a solved exercise, there is a point in the solution that I can't work out. I would be grateful if somebody could give me the detailed steps.
Consider the trivial principal bundle $P = M\times U(1)$ over a $C^\infty$-manifold $M$. Let $\Phi_t$ be the flow of a vector field $\mathfrak{X}(P)$.
Apparently, if $R_z$ designates the group action of $z \in U(1)$ on $M$, $X$ is $U(1)$-invariant ($R_z \cdot X=X$) if and only if $R_z$ commutes with $\Phi_t$ ($R_z \circ \Phi_t= \Phi_t \circ R_z$). Can somebody confirm this and help me with the proof ?
Thanks,
JD