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  "body": "\n## Definition\n\nLet $(M, g)$ be a [[Riemannian manifold|7CBT4SM83Y]], let $\\gamma \\colon [a, b] \\to M$ be a [[geodesic|NSYMQMHPDN]], and let $t_1 \u003c t_2$ in $[a, b]$.\nThe points $\\gamma(t_1)$ and $\\gamma(t_2)$ are *conjugate along* $\\gamma$ if there is a [[Jacobi field|9AHMM41PJ3]] $J$ along $\\gamma$, not identically zero, with $J(t_1) = 0$ and $J(t_2) = 0$.\nThe *multiplicity* of the pair is the dimension of the space of such Jacobi fields.\n\n## Remarks\n\nExample: on the unit sphere $S^n$, let $\\gamma(t)$ be a unit-speed great circle and $E$ a parallel unit field normal to $\\gamma'$.\nThe Jacobi field $J = \\sin(t) E$ vanishes at $t = 0$ and $t = \\pi$, so $\\gamma(0)$ and $\\gamma(\\pi)$, which are antipodal, are conjugate; there are $n - 1$ independent choices of $E$, so the multiplicity is $n - 1$.\n\nNon-example: in $\\R^n$ with the Euclidean metric, the Jacobi fields along $\\gamma(t) = p + tv$ are $J(t) = A + tB$ with constant vectors $A, B$.\nSuch a field vanishing at two times is zero, so no two points are conjugate.\n\nEquivalently, with $p = \\gamma(0)$ and $\\gamma(t) = \\exp_p(tv)$, the point $\\gamma(t_2)$ is conjugate to $p$ along $\\gamma$ if and only if the differential of the [[exponential map|Y7Z0AGY1A7]] $\\exp_p$ is singular at $t_2 v$.\nThe multiplicity is at most $n - 1$, where $n = \\dim M$.\n",
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  "name": "Conjugate points",
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