On the Blaschke's Conjecture

TL;DR

This paper proves Blaschke's conjecture for complete Riemannian manifolds with diameter and injectivity radius equal to pi/2, confirming the manifold's isometry to classical space forms under a sectional curvature bound.

Contribution

It establishes the validity of Blaschke's conjecture for manifolds with sectional curvature at least one, extending previous results.

Findings

Confirmed Blaschke's conjecture under sec_M ≥ 1

Identified the manifold as a classical space form

Provided a new condition for the conjecture's validity

Abstract

The Blaschke's conjecture asserts that if $\diam(M)=\text{Inj}(M)=\frac\pi2$ (up to a rescaling) for a complete Riemannian manifold $M$, then $M$ is isometric to $\Bbb S^n(\frac12)$, ${\Bbb R\Bbb P}^{n}$, ${\Bbb C\Bbb P}^{n}$, ${\Bbb H\Bbb P}^{n}$ or ${\Bbb Ca\Bbb P}^{2}$ endowed with the canonical metric. In the paper, we prove that the conjecture is true if we in addition assume that $\sec_M\geq1$.