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  "body": "\n## Definition\n\nLet $(M, g)$ be a complete connected [[Riemannian manifold|7CBT4SM83Y]] with [[Riemannian distance|B4QR306D8G]] $d$, and let $p \\in M$.\nFor a unit vector $v \\in T_pM$, let $\\gamma_v(t) = \\exp_p(tv)$ be the unit-speed [[geodesic|NSYMQMHPDN]] from $p$ in direction $v$, using the [[exponential map|Y7Z0AGY1A7]], and let\n$$t_c(v) = \\sup \\set{ t \u003e 0 : d(p, \\gamma_v(t)) = t } \\in (0, \\infty].$$\nIf $t_c(v) \u003c \\infty$, the point $\\gamma_v(t_c(v))$ is the *cut point* of $p$ along $\\gamma_v$.\nThe *cut locus* of $p$ is the set of all cut points of $p$, over all unit vectors $v \\in T_pM$.\n\n## Remarks\n\nExample: on the unit sphere $S^n$, every unit-speed geodesic from $p$ is a great circle with $d(p, \\gamma_v(t)) = t$ for $t \\le \\pi$ and $d(p, \\gamma_v(t)) \u003c t$ just after $t = \\pi$.\nSo $t_c(v) = \\pi$ for every $v$, and the cut locus of $p$ is the single point $-p$.\n\nNon-example: in $\\R^n$ with the Euclidean metric, $d(p, p + tv) = t$ for all $t \u003e 0$, so $t_c(v) = \\infty$ for all $v$, there are no cut points, and the cut locus is empty.\n\nLet $q = \\gamma_v(t_c(v))$ be the cut point of $p$ along $\\gamma_v$.\nThen $q$ is the first point [[conjugate|ZPXE5VSZ35]] to $p$ along $\\gamma_v$, or there are two distinct minimizing geodesics from $p$ to $q$, or both.\nThe cut locus is a closed set of measure zero, and $\\exp_p$ maps the open star-shaped set $\\set{tv : \\abs{v} = 1, \\ 0 \\le t \u003c t_c(v)}$ diffeomorphically onto $M$ minus the cut locus of $p$.\n",
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    "name": "Cut locus",
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