Besse projective spaces with many diameters (Q6174365)

A Besse manifold is a connected Riemannian manifold whose geodesics are periodic. A Blaschke manifold is a compact Riemannian manifold whose injectivity radius equals its diameter; it follows that its geodesics are periodic with length equal to twice the diameter. It is well known that \(\text{Blaschke}\subsetneq\text{Besse}\). The authors prove that Besse manifolds whose geodesics, in sufficiently many directions, are periodic with length equal to twice the diameter, are Blaschke. They also prove a theorem bounding the length of the shortest closed geodesic, on compact and simply connected Riemannian manifolds with pinched curvature, in terms of the pinching and the diameter.