Zoll metric on a Riemannian manifold is a metric for which all geodesics are closed and have the same period. For sure, a standart metric on the sphere $S^2$ has this property: all its geodesics are great circles of period $2 \pi$. Projective space $RP^2$ as a factor of $S^2$ provided with the canonical metric also has all geodesics closed and of the same lenght $\pi$. There was lots of work done (Tannery, Zoll, Funk, Guillemin and others) studying Zoll surfaces. For example, a theorem of Green shows that there are no nontrivial Zoll metrics on $RP^2$. On the contrary, there is an abundance of such metrics on the sphere $S^2$, even without nontrivial isometries.
My question is why Zoll metrics exist only on the sphere and its factor $RP^2$? Of course, here I restrict myself to the $2$-dimensional case. The evidence that is true is mentioned in the book A.Besse "Manifolds all of whose geodesics are closed". The style of the book is very formal and the statement is proven in such a generality that it's impossible to understand. There should be some easy topological argument but I do not find it.