Theorem (Michel). A $1$-form on the projective plane is exact if and only if its integrals over every projective line is equal to zero.

Is there a simple proof of this result due, I think, to R. Michel ?

I'm guessing there must be a representation theoretic proof (everything is $SL(3;\mathbb{R})$ equivariant) and a complex-geometric proof (where $\mathbb{R}P^2$ is complexified to $\mathbb{C}P^2$ ) and I would appreciate reference for these, but I'm really interested in a simple proof that one could teach to seniors or first year grads.

P.S. I have not been able to get a hold of Michel's papers. Perhaps everything is there and in that case I apologize before hand.

Theorem (Michel). A $1$-form on the projective plane is exact if and only if its integral over any projective line is equal to zero.

Is there a simple proof of this result due, I think, to R. Michel ?

I'm guessing there must be a representation theoretic proof (everything is $SL(3;\mathbb{R})$ equivariant) and a complex-geometric proof (where $\mathbb{R}P^2$ is complexified to $\mathbb{C}P^2$ ) and I would appreciate reference for these, but I'm really interested in a simple proof that one could teach to seniors or first year grads.

P.S. I have not been able to get a hold of Michel's papers. Perhaps everything is there and in that case I apologize before hand.