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  "body": "\n## Statement\n\nLet $(M, g)$ be a [[Riemannian manifold|7CBT4SM83Y]] and let $\\gamma \\colon [0, 1] \\to M$ be a [[geodesic|NSYMQMHPDN]] from $p = \\gamma(0)$ to $q = \\gamma(1)$.\nLet $V$ be the vector space of continuous, piecewise smooth vector fields $W$ along $\\gamma$ with $W(0) = 0$ and $W(1) = 0$.\nLet $I$ be the Hessian at $\\gamma$ of the [[energy functional on paths|Q12AB1MJ64]] from $p$ to $q$.\nIt is the symmetric bilinear form on $V$ given by\n$$I(W_1, W_2) = \\int_0^1 \\Big( g\\big(\\tfrac{D W_1}{dt}, \\tfrac{D W_2}{dt}\\big) - g\\big(R(W_1, \\dot\\gamma) \\dot\\gamma, W_2\\big) \\Big) \\, dt.$$\nHere $D/dt$ is the covariant derivative along $\\gamma$ and $R$ is the Riemann curvature tensor.\nThe sign is such that, for $W \\perp \\dot\\gamma$, $g(R(W, \\dot\\gamma)\\dot\\gamma, W)$ is $\\abs{W}^2 \\abs{\\dot\\gamma}^2$ times the sectional curvature of the plane of $W$ and $\\dot\\gamma$.\nThen:\n\n1. The [[Morse index|MAX2ZYG6H9]] of $I$, the largest dimension of a subspace of $V$ on which $I$ is negative definite, is finite.\n2. It equals the number of parameters $t \\in (0, 1)$ such that $\\gamma(t)$ is [[conjugate|ZPXE5VSZ35]] to $\\gamma(0)$ along $\\gamma$, each counted with its multiplicity.\n3. The nullity of $I$ equals the multiplicity of $q$ as a conjugate point of $p$ along $\\gamma$.\nIt is $0$ if $q$ is not conjugate to $p$.\n\n## Remarks\n\nIntegration by parts gives $I(W_1, W_2) = -\\int_0^1 g\\big(\\tfrac{D^2 W_1}{dt^2} + R(W_1, \\dot\\gamma)\\dot\\gamma, W_2\\big)\\,dt$ plus terms at the breaks of $W_1$, so the null space of $I$ consists of the [[Jacobi fields|9AHMM41PJ3]] vanishing at both ends.\nFor the index, one splits $[0, 1]$ into short intervals on which $\\gamma$ is minimizing.\nThen $V$ decomposes into the finite-dimensional space of broken Jacobi fields and a complement on which $I$ is positive definite.\nThen one tracks how the index of $I$ restricted to $[0, \\tau]$ changes as $\\tau$ grows: it jumps exactly at conjugate points, by their multiplicity.\n\nExample: on the unit sphere $S^n$, a unit-speed great circle arc of length $L$ with $\\pi \u003c L \u003c 2\\pi$ passes one conjugate point, the antipode of its start, with multiplicity $n - 1$.\nSo its index is $n - 1$, and for $n \\ge 2$ it is not a local minimum of the energy.\nIn particular, a geodesic that passes a conjugate point in its interior is not locally length-minimizing.\n\n## Explanations\n\n### L1 human\n\nA geodesic is a critical point of the energy among paths with the same ends, but it need not be a minimum.\nThe index counts the independent ways of shortening it by a small deformation.\nThe theorem says that this number is the number of conjugate points along the way, where nearby geodesics from the start refocus.\nOn a sphere, a great circle arc longer than half the circle passes the antipode of its start, and it can be shortened by sliding it over the sphere.\n\n### L2 human\n\nThe theorem turns a variational question into a question about an ordinary differential equation, the Jacobi equation.\nIt is the key input of Morse theory on path spaces: together with the Palais–Smale condition, it computes the cells attached at each geodesic.\nFor example, on a manifold of nonpositive sectional curvature there are no conjugate points, so every geodesic has index $0$.\nBott used it to compute indices of geodesics in compact Lie groups in his proof of periodicity.\n",
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    "msc": [
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  "msc": [
    "58E10",
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  "name": "Morse index theorem for geodesics",
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  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/YZ4/YZ4268SK8G.md",
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  "url": "https://tamnd.github.io/math2/wiki/morse-index-theorem-for-geodesics/",
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