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  "body": "\n## Statement\n\nLet $g$ be a smooth [[Riemannian metric|Q61VFHCH2J]] on the two-dimensional [[sphere|P9CM4GQ2T0]] $S^2$.\nThen $(S^2, g)$ has infinitely many geometrically distinct [[closed geodesics|4MCCP8GG4J]].\n\n## Remarks\n\nThe proof combines two results.\nFranks treated, with a theorem on area-preserving homeomorphisms of the open annulus, the metrics for which a simple closed geodesic gives an annulus of section for the geodesic flow.\nBangert treated the remaining metrics by variational methods on the loop space.\nThe [[theorem of the three geodesics|AB9FNZD18B]] gives a simple closed geodesic to start from.\n\nThe [[Gromoll–Meyer theorem|XC8BRYJN93]] does not apply here, because the Betti numbers of the free loop space of $S^2$ are bounded.\nThe hypothesis that the metric is Riemannian matters: Katok constructed non-reversible Finsler metrics on $S^2$ with only finitely many closed geodesics.\n\n## Explanations\n\n### L1 human\n\nHowever one bends a sphere, it carries infinitely many different closed loops that go straight ahead, though they may cross themselves.\nOn a round sphere this is clear, since every great circle is one.\nThe theorem says that no smooth deformation, however irregular, can destroy all but finitely many of them.\n\n### L2 human\n\nThe result answers a long-standing question for the simplest closed surface.\nIt mixes two methods: dynamics of the geodesic flow, through a surface of section and a return map of an annulus, and Morse theory of the energy on loop spaces.\nFor higher-dimensional spheres the corresponding question for Riemannian metrics is open.\n",
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