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  "body": "\n## Statement\n\nLet $(M, g)$ be a connected, simply connected, complete [[Riemannian manifold|7CBT4SM83Y]] of dimension $n \\ge 2$.\nSuppose that every [[sectional curvature|1X42JAA349]] $K$ of $M$ satisfies\n$$\\frac{1}{4} \u003c K \\le 1.$$\nThen:\n\n1. $M$ is [[homeomorphic|0QT00PB5G2]] to the [[sphere|T62369RFH6]] $S^n$ (Berger and Klingenberg).\n2. $M$ is [[diffeomorphic|0KZZ70ZHXP]] to $S^n$ (Brendle and Schoen).\n\n## Remarks\n\nBy Myers's theorem, $M$ is compact.\nThe idea of the topological proof: Klingenberg's estimate shows that the injectivity radius of $M$ is at least $\\pi$.\nWith Toponogov's comparison theorem one then shows that $M$ is the union of two embedded open balls centred at two points at maximal distance.\nA closed manifold that is the union of two embedded open discs is homeomorphic to a sphere.\nThe differentiable version uses the Ricci flow: Brendle and Schoen showed that the flow, normalized, takes the metric to one of constant curvature.\n\nThe 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 $1/4 \\le K \\le 1$.\nIt is simply connected and compact but not homeomorphic to a sphere.\nSimple connectedness is needed: real projective space $\\R P^n$ with the round metric has $K = 1$ and is not a sphere.\n\n## Explanations\n\n### L1 human\n\nSuppose a simply connected space bends positively.\nIf the bending in its most curved direction is less than four times that in its least curved direction, then the space has the shape of a sphere.\nNot only roughly but also smoothly: it can be deformed into a round sphere.\n\n### L2 human\n\nThe topological sphere theorem was proved by Berger and Klingenberg around 1960, after a version with a weaker constant by Rauch.\nThe smooth version, proved by Brendle and Schoen in 2009, answered the question whether exotic spheres can carry quarter-pinched metrics.\nTheir method also gives the result under a pointwise pinching condition.\n",
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