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  "body": "\n## Definition\n\nLet $(M, g)$ be a [[Riemannian manifold|7CBT4SM83Y]] and $p \\in M$.\nFor $v$ in the [[tangent space|WY6K3P4YCG]] $T_pM$, let $\\gamma_v$ be the maximal [[geodesic|NSYMQMHPDN]] with $\\gamma_v(0) = p$ and $\\gamma_v'(0) = v$.\nLet $\\mathcal{E}_p \\subseteq T_pM$ be the set of $v$ such that $\\gamma_v$ is defined on an interval containing $[0, 1]$.\nThe *exponential map* at $p$ is\n$$\\exp_p \\colon \\mathcal{E}_p \\to M, \\qquad \\exp_p(v) = \\gamma_v(1).$$\n\n## Remarks\n\nExample: in $\\R^n$ with the Euclidean metric, $\\gamma_v(t) = p + tv$, so $\\mathcal{E}_p = \\R^n$ and $\\exp_p(v) = p + v$.\n\nNon-example: in the punctured plane $M = \\R^2 \\setminus \\set{0}$ with $p = (1, 0)$ and $v = (-2, 0)$, the geodesic $\\gamma_v(t) = (1 - 2t, 0)$ hits the removed origin at $t = 1/2$.\nSo $v \\notin \\mathcal{E}_p$, and $\\exp_p(v)$ is not defined.\n\nThe set $\\mathcal{E}_p$ is open and star-shaped about $0$, and $\\exp_p$ is smooth on it.\nBy rescaling, $\\gamma_v(t) = \\exp_p(tv)$.\nThe differential of $\\exp_p$ at $0$ is the identity of $T_pM$.\nSo, by the inverse function theorem, $\\exp_p$ maps a neighbourhood of $0$ diffeomorphically onto a neighbourhood of $p$.\n",
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