**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. 



Forms satisfying the zero-energy condition on the projective plane 
There is a simple proof along the following lines: Because the first deRham cohomology group of $\mathbb{RP}^2$ is trivial, a $1$-form on $\mathbb{RP}^2$ is exact if and only if it is closed. Let $\alpha$ be a $1$-form on $\mathbb{RP}^2$ whose integral over every line vanishes, and let $\beta = \pi^\*\alpha$ where $\pi:S^2\to \mathbb{RP}^2$ is the standard double cover. Then the integral of $\beta$ over every great circle vanishes and $\beta$ is invariant under the antipodal involution $\iota:S^2\to S^2$, so $d\beta$ is a $2$-form on $S^2$ that is $\iota$-invariant and its integral over every hemisphere must vanish. 


Now $d\beta = b\ dA$ where $dA$ is the standard volume form on $S^2$ and we must have $b\circ\iota=-b$ in order for $d\beta$ to be invariant. Now you use the representation theoretic fact that the operation $A:C^\infty(S^2)\to C^\infty(S^2)$ defined by
$$
Af(u) = \frac1{2\pi}\int\_{v\cdot u\ge0} f\ dA
$$
for $u\in S^2$ is $\mathrm{SO}(3)$-equivariant and hence must, on each eigenspace of the Laplacian on $S^2$, be a multiple of the identity (since these eigenspaces are irreducible representations of $\mathrm{SO}(3)$). To complete the proof that $Ab=0$ implies $b=0$ (and hence that $d\beta=0$), one just needs to check that $A$ is nonzero on each $\iota$-odd eigenspace of the Laplacian, and this is straightforward (see below for one proof of this).


*Added comment:* Here's a simple way to see that $A$ is nonzero on the odd eigenspaces (without having to do any explicit integration). The $d$-th eigenspace $H\_d$ of the Laplacian on $S^2$ is simply the restriction to $S^2$ of the harmonic polynomials on $\mathbb{R}^3$ that are homogeneous of degree $d$. Since $A$ acts as a multiple of the identity on each of these spaces, it's enough to show that, when $d=2m{+}1$, $A$ is nonzero on at least one element of $H\_d$. To do this, let $x$, $y$, and $z$ be the standard coordinates on $\mathbb{R}^3$ and consider the polynomial
$$
f = \mathrm{Re}\bigl((x+i y)^{2m+1}\bigr) 
= \prod\_{k=-m}^m \left( x-\tan\left(\frac{k\pi}{2m{+}1}\right) y\right).
$$
Clearly, $f$ belongs to $H\_{2m+1}$ and $f(1,0,0)=1\not=0$. Moreover, $f$ is odd with respect to rotation by an angle of $\pi/(2m{+}1)$ about the $z$-axis, and it vanishes on the $2m{+}1$ planes given by the above factors, which divide the sphere into $4m{+}2$ congruent 'sectors' or 'spherical wedges', on half of which $f$ is positive and on half of which $f$ is negative. Now, $Af(1,0,0)$ is the average of $f$ over the hemisphere on which $x\ge0$, and this is composed of $2m{+}1$ of these sectors, which alternate in the sign of $f$. Since there is an odd number of them, their integrals can't cancel out, so $Af(1,0,0)$ cannot be zero. Thus, $A$ is nonzero on the odd harmonics, as was to be shown.


