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  "body": "\n## Statement\n\nLet $(M, g)$ be a compact [[Riemannian manifold|7CBT4SM83Y]] without boundary, of dimension at least $1$.\nThen $M$ has a [[closed geodesic|4MCCP8GG4J]].\n\n## Remarks\n\nIf the fundamental group of $M$ is nontrivial, a loop of least length in a nontrivial free homotopy class exists by compactness, and it is a closed geodesic.\nIf $M$ is simply connected, then $M$ is not contractible, because a closed $n$-manifold has nonzero homology in degree $n$ with coefficients in $\\Z/2$.\nBy the Hurewicz theorem, some homotopy group $\\pi_k(M)$ with $k \\ge 2$ is nonzero.\nThen a minimax over $(k-1)$-parameter families of loops in the [[free loop space|SGGXZ3TBSA]] gives a critical value of the energy.\nThe Palais–Smale condition for the energy makes the minimax work.\nThe nontriviality of the class makes the critical value positive, so the critical point is not a constant loop.\n\nExample: on a round sphere every great circle is a closed geodesic.\nOn an ellipsoid in $\\R^3$ the theorem gives at least one, and the intersections with the three coordinate planes are in fact closed geodesics.\nCompactness is needed: in $\\R^n$ with the Euclidean metric no geodesic is closed.\n\n## Explanations\n\n### L1 human\n\nOn any closed curved space, there is a loop that runs straight ahead and closes up on itself.\nIf the space has a hole that a loop can wind around, one can tighten such a loop as much as possible.\nIf not, the space still has a higher-dimensional hole.\nOne sweeps it by a family of loops and looks for the loop at the highest point of the best sweep.\n\n### L2 human\n\nThe theorem was the first general existence result for closed geodesics, and its proof is the model for minimax methods on loop spaces.\nIt extends Birkhoff's earlier result for spheres, which uses the [[Birkhoff min-max method|1R5X9HW6D3]].\nQuestions about the number of geometrically distinct closed geodesics are much harder; see the [[Gromoll–Meyer theorem|XC8BRYJN93]] and the [[theorem of Bangert and Franks|X93AEE88PX]].\n",
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