Real projective spaces with all geodesics closed

Abstract

Let M be a smooth n-manifold admitting an even degree smooth covering by a smooth homotopy n-sphere. When \({n \neq 3}\), we prove that if g is a Riemannian metric on M with all geodesics closed, then g has constant sectional curvatures. In particular, n-dimensional real projective spaces (\({n \neq 3}\)) with all geodesics closed have constant sectional curvatures.