It is known that the cohomology ring of a Zoll manifold---a riemannian manifold all of whose geodesics are periodic with the same minimal period---must be the same as the cohomology ring of a compact rank one symmetric space (see Besse's book Manifolds all of whose geodesics are closed for references).

Is there a simple and elementary proof of the following much weaker property?

The first Betti number of a Zoll manifold is equal to zero.