Zoll manifold classification by compact rank-one symmetric spaces

Determine whether every Zoll manifold is diffeomorphic to a compact rank-one symmetric space.

Background

The paper places hypersurface Zoll manifolds in the broader context of the topology of ordinary Zoll manifolds. Although Bott–Samelson theory shows that a manifold admitting a Zoll metric has the integral cohomology ring of a compact rank-one symmetric space, the paper notes that no example with a different topology is known. The unresolved issue is whether this cohomological restriction can be strengthened to a diffeomorphism classification by compact rank-one symmetric spaces.