On the Blaschke's Conjecture

The Blaschke’s conjecture asserts that if $\text{diam}(M)=\text{Inj}(M)=\frac{\pi }{2}$ (up to a rescaling) for a complete Riemannian manifold $M$, then $M$ is isometric to ${\mathbb{S}}^n(\frac{1}{2})$, ${\mathbb{R}\mathbb{P}}^n$, ${\mathbb{C}\mathbb{P}}^n$, ${\mathbb{H}\mathbb{P}}^n$ or ${\mathbb{C}a\mathbb{P}}^2$ endowed with the canonical metric. In the paper, we prove that the conjecture is true if we in addition assume that ${\sec }_M\ge 1$.