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  "body": "\n## Definition\n\nLet $(M, g)$ be a [[Riemannian manifold|7CBT4SM83Y]] with Levi-Civita connection $\\nabla$ and curvature $R(X, Y)Z = \\nabla_X \\nabla_Y Z - \\nabla_Y \\nabla_X Z - \\nabla_{[X, Y]} Z$, see [[Riemann curvature tensor|JA29ETCM02]].\nLet $\\gamma \\colon I \\to M$ be a [[geodesic|NSYMQMHPDN]], and let $D_t$ be the [[covariant derivative|5FZDQQHQKM]] along $\\gamma$.\nA smooth vector field $J$ along $\\gamma$ is a *Jacobi field* if\n$$D_t^2 J + R(J, \\gamma')\\gamma' = 0 \\quad \\text{on } I.$$\n\n## Remarks\n\nExample: on the unit sphere $S^2$, let $\\gamma$ be a unit-speed geodesic and $E$ a parallel unit field along $\\gamma$ orthogonal to $\\gamma'$.\nFor $J = \\sin(t) E$, one has $D_t^2 J = -\\sin(t) E$ and, since the curvature is $1$, $R(J, \\gamma')\\gamma' = \\sin(t) E$; the sum is $0$, so $J$ is a Jacobi field.\n\nNon-example: on the same sphere, $J = tE$ has $D_t^2 J = 0$ but $R(J, \\gamma')\\gamma' = tE \\ne 0$ for $t \\ne 0$, so $J$ is not a Jacobi field, although it is one in the flat plane.\n\nIf $\\Gamma(s, t)$ is a smooth family of geodesics $t \\mapsto \\Gamma(s, t)$, then $J(t) = \\partial_s \\Gamma(0, t)$ is a Jacobi field, and every Jacobi field along a geodesic on a compact interval arises in this way.\nA Jacobi field is determined by $J(t_0)$ and $D_t J(t_0)$ at one time, so the Jacobi fields along $\\gamma$ form a vector space of dimension $2n$, where $n = \\dim M$.\nThe fields $\\gamma'$ and $t \\gamma'$ are always Jacobi fields; the interesting ones are normal to $\\gamma$.\nWith the opposite sign convention for $R$, the equation reads $D_t^2 J + R(\\gamma', J)\\gamma' = 0$.\n",
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