On Blaschke's conjecture

Xiaole Su, Hongwei Sun and Yusheng Wang

Vol. 282 (2016), No. 2, 479–485

DOI: 10.2140/pjm.2016.282.479

Abstract

Blaschke’s conjecture asserts that if a complete Riemannian manifold $M$ satisfies $diam\left(M\right)=Inj\left(M\right)=\frac{\pi }{2}$, then $M$ is isometric to ${\mathbb{S}}^n\left(\frac{1}{2}\right)$ or to the real, complex, quaternionic or octonionic projective plane with its canonical metric. We prove that the conjecture is true under the assumption that ${sec}_M\ge 1$.

Keywords

Blaschke's conjecture, Berger's rigidity theorem, Toponogov's comparison theorem

Mathematical Subject Classification 2010

Primary: 53C20

Milestones

Received: 11 June 2015

Accepted: 19 December 2015

Published: 3 March 2016