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  "body": "\n## Definition\n\nLet $(M, g)$ be a connected [[Riemannian manifold|7CBT4SM83Y]] with [[Riemannian metric|Q61VFHCH2J]] $g$.\nFor a piecewise smooth curve $\\gamma \\colon [a, b] \\to M$, its [[length|0TH78S7GBZ]] is\n$$L(\\gamma) = \\int_a^b \\sqrt{g_{\\gamma(t)}(\\gamma'(t), \\gamma'(t))} \\, dt.$$\nThe *Riemannian distance* between points $p, q \\in M$ is the [[infimum|0QF1MZ9FRG]]\n$$d_g(p, q) = \\inf \\set{ L(\\gamma) : \\gamma \\text{ is a piecewise smooth curve from } p \\text{ to } q }.$$\n\n## Remarks\n\nExample: on $\\R^n$ with the Euclidean metric $g = \\sum_i dx_i^2$, the straight segment from $p$ to $q$ has length $\\abs{q - p}$, and no curve is shorter, so $d_g(p, q) = \\abs{q - p}$.\n\nNon-example: if $M$ is not connected, two points in different components are joined by no curve.\nThen the infimum is over the empty set, and $d_g$ is not a finite distance.\nSome authors then set $d_g(p, q) = \\infty$.\n\nThe function $d_g$ is a [[metric|0VKD1TP7F3]] on $M$, and the topology it induces is the given topology of $M$.\nPositivity $d_g(p, q) \u003e 0$ for $p \\ne q$ is the nontrivial part; it follows by comparing $g$ with a Euclidean metric in a chart.\nThe infimum need not be attained: in $\\R^2 \\setminus \\set{0}$ with the Euclidean metric, $d_g((-1, 0), (1, 0)) = 2$, but every curve between these points avoiding $0$ is longer than $2$.\n",
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