Sphere theorem

Remarks

By Myers's theorem, $M$ is compact. The idea of the topological proof: Klingenberg's estimate shows that the injectivity radius of $M$ is at least $\pi$. With Toponogov's comparison theorem one then shows that $M$ is the union of two embedded open balls centred at two points at maximal distance. A closed manifold that is the union of two embedded open discs is homeomorphic to a sphere. The differentiable version uses the Ricci flow: Brendle and Schoen showed that the flow, normalized, takes the metric to one of constant curvature.

The constant $1/4$ is sharp: take the complex projective space $\C P^m$ with $m \ge 2$ and the Fubini–Study metric scaled so that $1/4 \le K \le 1$. It is simply connected and compact but not homeomorphic to a sphere. Simple connectedness is needed: real projective space $\R P^n$ with the round metric has $K = 1$ and is not a sphere.

Going deeper

For a student

The topological sphere theorem was proved by Berger and Klingenberg around 1960, after a version with a weaker constant by Rauch. The smooth version, proved by Brendle and Schoen in 2009, answered the question whether exotic spheres can carry quarter-pinched metrics. Their method also gives the result under a pointwise pinching condition.