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  "body": "\n## Definition\n\nLet $(M, g)$ be a [[pseudo-Riemannian manifold|GH6CXWFNDJ]] with [[Levi-Civita connection|CAZYGE1T4E]] $\\nabla$, and let $I \\subseteq \\R$ be an interval.\nA smooth curve $\\gamma \\colon I \\to M$ is a *geodesic* if\n$$D_t \\gamma'(t) = 0 \\quad \\text{for all } t \\in I,$$\nwhere $D_t$ is the [[covariant derivative|5FZDQQHQKM]] along $\\gamma$ induced by $\\nabla$.\n\n## Remarks\n\nIn local coordinates, with Christoffel symbols $\\Gamma^k_{ij}$, the condition is the geodesic equation\n$$\\ddot\\gamma^k + \\sum_{i,j} \\Gamma^k_{ij}(\\gamma) \\, \\dot\\gamma^i \\dot\\gamma^j = 0, \\qquad k = 1, \\ldots, n.$$\n\nExample: in $\\R^n$ with the Euclidean metric all $\\Gamma^k_{ij}$ vanish, so the geodesics are the curves with $\\ddot\\gamma = 0$, that is $\\gamma(t) = p + tv$.\n\nNon-example: in $\\R^2$, the curve $\\gamma(t) = (t^2, 0)$ traces a straight line, but $\\ddot\\gamma = (2, 0) \\ne 0$, so it is not a geodesic: a geodesic must have constant speed.\n\nThe quantity $g(\\gamma', \\gamma')$ is constant along a geodesic.\nOn a Riemannian manifold, every geodesic is locally length-minimizing: each $t \\in I$ has a neighbourhood in $I$ on which $\\gamma$ is a shortest curve between its endpoints.\nFor each $p \\in M$ and $v \\in T_pM$ there is a unique maximal geodesic $\\gamma_v$ with $\\gamma_v(0) = p$ and $\\gamma_v'(0) = v$.\n",
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    "name": "Geodesic",
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  "msc": [
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  "name": "Geodesic",
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