The Blaschke manifold classification conjecture

Let $M$ be a Riemannian manifold. Its injectivity radius $\operatorname{inj} M$ is the infimum of the injectivity radii at its points, and its diameter $\operatorname{diam} M$ is the supremum of the distances between points of $M$. A manifold satisfying

$$\operatorname{inj} M = \operatorname{diam} M$$

is called a Blaschke manifold. A compact rank-one symmetric space, abbreviated CROSS, is a compact Riemannian symmetric space of rank one.

Blaschke manifold classification conjecture. If $M$ is a Riemannian manifold such that

$$\operatorname{inj} M = \operatorname{diam} M,$$

then $M$ is isometric to a compact rank-one symmetric space (CROSS).

The conjecture asks whether the metric condition defining Blaschke manifolds forces the standard rank-one symmetric-space geometry. It is presented as an open problem in the source, and the paper studies this question in connection with spherical rank rigidity.