On Blaschke's conjecture (Q265541)

Blaschke's conjecture asserts that if a complete Riemannian \(n\)-manifold \(M\) satisfies \(\mathrm{diam}(M) =\mathrm{inj}(M) = \frac{\pi}{2}\), then \(M\) is isometric to \(S^n \left(\frac{1}{2}\right)\) or to the real, complex, quaternionic or octonionic projective space with its canonical metric. When the sectional curvature \(K_M\) satisfies \(K_M \leq 4\), Blaschke's conjecture holds due to [\textit{V. Rovenskii} and \textit{V. Toponogov}, ``Great sphere foliations and manifolds with curvature bounded above'', in: Foliations on Riemannian manifolds and submanifold. Boston: Birkhäuser (1998; Zbl 0958.53021), Appendix A, 218--234]. See also [\textit{K. Shankar} et al., Duke Math. J. 128, No. 1, 65--81 (2005; Zbl 1082.53051)].NEWLINENEWLINE NEWLINEIn this paper, the authors prove that Blaschke's conjecture is true when \(K_M \geq 1\). First, from the proof of Berger's rigidity theorem in [\textit{J. Cheeger} and \textit{D. G. Ebin}, Comparison theorems in Riemannian geometry. Amsterdam-Oxford: North-Holland Publishing Company; New York: American Elsevier Publishing Company, Inc (1975; Zbl 0309.53035)], the authors observe that if \(M\) is a complete Riemannian manifold with \(1 \leq K_M \leq 4\) and \(\mathrm{diam}(M) = \mathrm{inj}(M) = \frac{\pi}{2}\), then \(M\) is isometric to \(S^n \left(\frac{1}{2}\right)\) or a \(\mathbb {KP}^n\), where \(\mathbb K\) is one of the division algebras of real, complex, quaternionic or octonionic numbers. As a second ingredient for the proof of the result, the authors show that for any point \(p \in M\), the set \(\{q\in M: d(p, q) = \frac{\pi}{2}\}\) is a complete, totally geodesic submanifold in \(M\). Using these together with Toponogov comparison theorem, the authors conclude the main result.