Geodesics Closed On The Projective Plane

Abstract.

Given a C ^∞ Riemannian metric g on ${\mathbb{R}}$ P ² we prove that (${\mathbb{R}}P^2$, g) has constant curvature iff all geodesics are closed. Therefore ${\mathbb{R}}P^2$ 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.