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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 at least three geometrically distinct simple [[closed geodesics|4MCCP8GG4J]], where simple means without self-intersections.\n\n## Remarks\n\nThe idea goes back to Lyusternik and Schnirelmann.\nConsider the space of unoriented simple closed curves on $S^2$, together with the point curves.\nIts topology is rich enough to force three minimax levels, in the spirit of [[Lusternik–Schnirelmann theory|QARH15D2PN]].\nThe difficulty is to deform curves by a length-decreasing process that keeps them simple; this caused gaps in the early arguments.\nOne complete proof uses the [[curve-shortening flow|DVGFRRYK10]], which preserves simplicity and, by a theorem of Grayson, either shrinks a curve to a point or converges to a closed geodesic.\n\nExample: on a round sphere every great circle is a simple closed geodesic, so there are infinitely many.\nOn an ellipsoid in $\\R^3$ with three distinct axes, the three principal sections in the coordinate planes are simple closed geodesics.\nThe bound three cannot be improved in general.\n\n## Explanations\n\n### L1 human\n\nHowever one deforms a sphere smoothly, it still carries at least three different closed loops that go straight ahead and never cross themselves.\nOn a round sphere there are infinitely many such loops, the great circles.\nOn an egg-shaped surface with three different axes there are typically exactly three, the waists around the three axes.\n\n### L2 human\n\nThe theorem is a sharp lower bound for simple closed geodesics.\nWhen self-intersections are allowed, the theorem of [[Bangert and Franks|X93AEE88PX]] gives infinitely many closed geodesics on any Riemannian $S^2$.\nIts proof is a model of Lusternik–Schnirelmann theory on a space of curves: three independent families of curves sweep out the sphere, and each family yields a critical level.\nThe first step, a single closed geodesic on $S^2$, is the [[Birkhoff min-max method|1R5X9HW6D3]].\n",
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