# Can every De Rham cohomology class be represented by a closed form $\alpha$ with $L_X \alpha=0$

Assume that $M$ is a manifold and $X$ is a vector field on $M$.

Is it true to say that every closed form is De Rham-cohomologue to a closed form $\alpha$ with $L_X \alpha =0$?

• What did you try already? – Vladimir Dotsenko Jun 26 '18 at 8:07
• @VladimirDotsenko For open orientable manifolds the question has obviously positive answer in their last cohomology. So I am curious if the question has trivial answer in all cases, compact or non compact, in all cohomology dimension. – Ali Taghavi Jun 26 '18 at 8:25

You could try the vector field $X=x\partial_x$ on the real projective line, so with affine coordinate $x$. Then in another affine chart $y=1/x$, $X=-y\partial_y$, so $X$ is smooth everywhere. Every 1-form on the real projective line is closed. An invariant 1-form has to be $\alpha=C dx/x$. To be defined near $x=0$, it has to be $0$. But on the real projective line, there is cohomology in dimension 1, as the real projective line is the circle.
• If a compact Lie group $G$ acts smoothly on a manifold $M$, then every smooth closed form on $M$ is cohomologous to a $G$ invariant closed form, by averaging over the biinvariant volume form on $G$. So there are interesting examples, like the circle with a rotation vector field. – Ben McKay Jun 26 '18 at 9:53