2 00 7 If all geodesics are closed on the projective plane

Given a C∞ Riemannian metric g on RP 2 we prove that (RP , g) has constant curvature iff all geodesics are closed. Therefore RP 2 is the first non trivial example of a manifold such that the smooth Riemannian metrics which involve that all geodesics are closed are unique up to isometries and scaling. This remarkable phenomenon is not true on the 2-sphere, since there is a large set of C∞ metrics whose geodesics are all closed and have the same period 2π (called Zoll metrics), but no metric of this set can be obtained from another metric of this set via an isometry and scaling. As a corollary we conclude that all two dimensional P-manifolds are SC-manifolds. 0 Introduction Gromoll and Grove proved in [G] that if (S, g) is a P-manifold then it is a SC-manifold (see defintion below). We prove that this even holds if RP 2 is a P-manifold. Moreover we show that RP 2 has constant curvature iff all geodesics are closed. The first result in this direction was proven by Green in [Gr]. He proved that S has constant curvature iff it is a Blaschke manifold. From this theorem it follows that (RP , g) has constant curvature iff all geodesics are closed, have the same period and are without selfintersections, since the orientable double cover of (RP , g) is then a Blaschke manifold. In the complete paper all geodesics are parametrized by arc length and the geodesic flow is complete. The manifolds and the Riemannian metrics are C. π : TM → M denotes the canonical projection. γv denotes the geodesic with γ̇v(0) = v. For interested readers who want to know more about P-manifolds we refer to the book [Bs].