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  "body": "\n## Statement\n\nLet $(M, g)$ be a connected complete [[Riemannian manifold|7CBT4SM83Y]] of dimension $n \\ge 2$.\nSuppose there is a real number $k \u003e 0$ such that the [[Ricci curvature|4GTNM0WHDH]] satisfies $\\operatorname{Ric}(v, v) \\ge (n-1) k \\, g(v, v)$ for every tangent vector $v$.\nThen:\n\n1. The [[diameter|0KW3PN547K]] of $M$ for the [[Riemannian distance|B4QR306D8G]] is at most $\\pi / \\sqrt{k}$.\n2. $M$ is compact.\n3. The [[fundamental group|S1XK3HXHFX]] of $M$ is finite.\n\n## Remarks\n\nThe idea of the proof: let $\\gamma$ be a minimizing unit-speed geodesic of length $L \u003e \\pi/\\sqrt{k}$.\nTake an orthonormal frame $E_1, \\dots, E_{n-1}$ of parallel fields normal to $\\gamma$ and the variation fields $\\sin(\\pi t / L) E_i$.\nBy the [[second variation formula|ZB91TH6FKV]], the sum of the second variations of length is negative, so one of them is negative and $\\gamma$ is not minimizing, a contradiction.\nPart 2 follows from part 1 and the Hopf–Rinow theorem.\nFor part 3, apply parts 1 and 2 to the [[universal cover|Y1YPE6PM52]] with the pulled-back metric: it is compact, so the covering has finitely many sheets.\n\nExample: the unit sphere $S^n$ has $\\operatorname{Ric} = (n-1) g$ and diameter exactly $\\pi$, so the bound is sharp.\nBy a theorem of Cheng, equality of the diameter with $\\pi/\\sqrt{k}$ holds only for the round sphere of radius $1/\\sqrt{k}$.\n\nThe uniform lower bound is needed: the paraboloid $z = x^2 + y^2$ in $\\R^3$ is complete with positive curvature, but its curvature tends to $0$ at infinity and it is not compact.\n\n## Explanations\n\n### L1 human\n\nIf a space is curved positively at least by a fixed amount in every direction, it must close up on itself, like a sphere.\nThe theorem gives an explicit bound: no two points are farther apart than on the sphere of the same curvature.\n\n### L2 human\n\nMyers's theorem is one of the first results linking a lower bound on Ricci curvature to global topology.\nFor example, it shows that a compact manifold with infinite fundamental group, such as a torus, carries no metric of positive Ricci curvature.\nIt has a sharp volume analogue, the Bishop–Gromov inequality, and it is used in general relativity to show that certain spacetimes are closed.\n",
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