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  "body": "\n## Definition\n\nLet $(M, g)$ be a [[Riemannian manifold|7CBT4SM83Y]] of dimension $n$ and $p \\in M$.\nLet $V \\subseteq T_pM$ be an open neighbourhood of $0$, star-shaped about $0$.\nAssume that the [[exponential map|Y7Z0AGY1A7]] $\\exp_p$ restricts to a [[diffeomorphism|0KZZ70ZHXP]] from $V$ onto an open set $U \\subseteq M$.\nLet $(e_1, \\ldots, e_n)$ be an [[orthonormal basis|0FWJ0VW561]] of $T_pM$ for $g_p$, and let $E \\colon \\R^n \\to T_pM$ be $E(x) = \\sum_i x^i e_i$.\nThe *normal coordinates* centred at $p$ are the chart\n$$\\varphi = E^{-1} \\circ (\\exp_p|_V)^{-1} \\colon U \\to \\R^n.$$\n\n## Remarks\n\nExample: in $\\R^n$ with the Euclidean metric and the standard basis, $\\exp_p(v) = p + v$, so the normal coordinates centred at $p$ are $\\varphi(q) = q - p$, defined on all of $\\R^n$.\n\nNon-example: in $\\R^2$ with the Euclidean metric, the chart $\\psi(q) = 2(q - p)$ has the form $E^{-1} \\circ \\exp_p^{-1}$ only for the basis $e_i = \\tfrac{1}{2} \\varepsilon_i$, where $\\varepsilon_1, \\varepsilon_2$ is the standard basis.\nThis basis is orthogonal but not orthonormal, so $\\psi$ is not a normal coordinate chart; indeed, in $\\psi$ the metric has matrix $\\tfrac{1}{4} I$ at $p$, not $I$.\n\nIn normal coordinates centred at $p$: $g_{ij}(p) = \\delta_{ij}$, $\\partial_k g_{ij}(p) = 0$ and $\\Gamma^k_{ij}(p) = 0$.\nThe geodesics through $p$ are the lines $t \\mapsto t x$ in these coordinates.\nDifferent orthonormal bases give normal coordinates that differ by an orthogonal matrix.\n",
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