The Blaschke conjecture: Blaschke manifolds are compact rank-one symmetric spaces

Theorem 1.1: if $(M,g)$ is connected, closed, smooth, of positive dimension, with $\mathrm{inj}(M,g)=\mathrm{diam}(M,g)$, then $(M,g)$ is isometric up to scale to a round sphere, a standard real, complex or quaternionic projective space, or the Cayley plane. Theorem 1.2 gives $\mathrm{Vol}(M,g)\ge\mathrm{Vol}(M_0,g_0)$ against the model at equal diameter, with equality only for isometry; the upper bound in quaternionic dimension $4r$, $r\ge2$, is proved from characteristic classes and the geodesic space. It does not treat the weaker Besse-type questions on manifolds all of whose geodesics are closed without the Blaschke condition (SC or P-manifolds in general).

Theorem 1.1: if $(M,g)$ is connected, closed, smooth, of positive dimension, with $\mathrm{inj}(M,g)=\mathrm{diam}(M,g)$, then $(M,g)$ is isometric up to scale to a round sphere, a standard real, complex or quaternionic projective space, or the Cayley plane. Theorem 1.2 gives $\mathrm{Vol}(M,g)\ge\mathrm{Vol}(M_0,g_0)$ against the model at equal diameter, with equality only for isometry; the upper bound in quaternionic dimension $4r$, $r\ge2$, is proved from characteristic classes and the geodesic space. It does not treat the weaker Besse-type questions on manifolds all of whose geodesics are closed without the Blaschke condition (SC or P-manifolds in general).