Summary of progress on the Blaschke conjecture

TL;DR

This paper reviews progress on the Blaschke conjecture, discussing the proof that manifolds with equal injectivity radius and diameter resemble rank one symmetric spaces, including special cases and topological classifications.

Contribution

It summarizes key proofs and known results regarding the topology and geometry of manifolds satisfying the Blaschke conjecture, including special cases like homology spheres.

Findings

Manifolds with equal injectivity radius and diameter have the cohomology of rank one symmetric spaces.

The conjecture is proven for homology spheres and real projective spaces.

Current knowledge on the diffeomorphism, homeomorphism, and homotopy types is summarized.

Abstract

The Blaschke conjecture claims that every compact Riemannian manifold whose injectivity radius equals its diameter is, up to constant rescaling, a compact rank one symmetric space. We summarize the intuition behind this problem, the proof that such manifolds have the cohomology of compact rank one symmetric spaces, and the proof of the conjecture for homology spheres and homology real projective spaces. We also summarize what is known on the diffeomorphism, homeomorphism and homotopy types of such manifolds.