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  "body": "\n## Definition\n\nLet $(M, g)$ be a [[Riemannian manifold|7CBT4SM83Y]].\nA *closed geodesic* is a nonconstant [[geodesic|NSYMQMHPDN]] $\\gamma \\colon \\R \\to M$ that is periodic: there is $T \u003e 0$ with $\\gamma(t + T) = \\gamma(t)$ for all $t \\in \\R$.\nTwo closed geodesics are *geometrically distinct* if their images are different sets.\n\n## Remarks\n\nEquivalently, a closed geodesic is a smooth map $\\gamma \\colon \\R / T\\Z \\to M$ that satisfies the geodesic equation and is not constant.\nA geodesic $\\gamma$ is closed if and only if $\\gamma(T) = \\gamma(0)$ and $\\dot\\gamma(T) = \\dot\\gamma(0)$ for some $T \u003e 0$, by uniqueness of geodesics with given initial point and velocity.\nThe closed geodesics are the critical points with positive energy of the energy functional $E(\\gamma) = \\frac{1}{2} \\int_0^1 \\abs{\\dot\\gamma}^2 \\, dt$ on the [[free loop space|SGGXZ3TBSA]] of $M$.\nEach closed geodesic of period $T$ gives infinitely many critical points, its iterates $t \\mapsto \\gamma(kt)$ for $k \\ge 1$, which is why counting geometrically distinct ones is hard.\n\nExample: on the unit sphere $S^2 \\subset \\R^3$, the great circle $\\gamma(t) = (\\cos t, \\sin t, 0)$ is a geodesic with period $2\\pi$, so it is a closed geodesic.\n\nNon-example: on the flat torus $\\R^2 / \\Z^2$, the geodesic $\\gamma(t) = (t, \\sqrt{2}\\, t) \\bmod \\Z^2$ is not periodic.\nIndeed $\\gamma(t + T) = \\gamma(t)$ would need $T \\in \\Z$ and $\\sqrt{2}\\, T \\in \\Z$, which is impossible for $T \u003e 0$.\n",
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    "name": "Closed geodesic",
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        "url": "https://en.wikipedia.org/wiki/Closed_geodesic"
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