Geodesics closed on the projective plane (Q1005065)

The author proves that the real projective plane equipped with a (smooth) Riemannian metric has constant curvature, if and only if, all of its geodesics are closed. Thus \(\mathbb{R}\mathbb{P}^2\) is the first non-trivial example of a manifold such that the Riemannian metrics having the property that all geodesics are closed are unique up to isometries and scaling. As a corollary, it is obtained that all two-dimensional P-manifolds (i.e., Riemannian manifolds all of whose geodesics are closed) are SC-manifolds (i.e., P-manifolds whose geodesics are simple closed, and all of them have the same period).