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  "body": "\n## Definition\n\nLet $M$ be a smooth manifold of dimension $n$ with an [[affine connection|JX65KSVHSJ]] $\\nabla$.\nLet $(U, x^1, \\dots, x^n)$ be a chart, and let $\\partial_i = \\partial / \\partial x^i$ be the coordinate vector fields on $U$.\nThe *Christoffel symbols* of $\\nabla$ in this chart are the $n^3$ smooth functions $\\Gamma^k_{ij}$ on $U$ determined by\n$$\\nabla_{\\partial_i} \\partial_j = \\sum_{k=1}^n \\Gamma^k_{ij} \\, \\partial_k \\qquad (1 \\le i, j \\le n).$$\n\n## Remarks\n\nThey determine $\\nabla$ on $U$: $\\nabla_X Y = \\sum_k \\big( X Y^k + \\sum_{i,j} \\Gamma^k_{ij} X^i Y^j \\big) \\partial_k$ for $X = \\sum X^i \\partial_i$ and $Y = \\sum Y^j \\partial_j$.\nFor the Levi-Civita connection of a [[Riemannian metric|Q61VFHCH2J]] with components $g_{ij}$ and inverse matrix $(g^{kl})$,\n$$\\Gamma^k_{ij} = \\tfrac12 \\sum_l g^{kl} \\big( \\partial_i g_{jl} + \\partial_j g_{il} - \\partial_l g_{ij} \\big).$$\nExample: for the flat connection of $\\R^2$ in polar coordinates $(r, \\theta)$, where $g = dr^2 + r^2 d\\theta^2$, this formula gives $\\Gamma^r_{\\theta\\theta} = -r$, $\\Gamma^\\theta_{r\\theta} = \\Gamma^\\theta_{\\theta r} = 1/r$, and all other symbols vanish.\nNon-example: the $\\Gamma^k_{ij}$ are not the components of a tensor field; under a change of chart they transform with an extra term containing second derivatives of the coordinate change.\nSo they can vanish at a point in one chart and not in another; in Cartesian coordinates the symbols above are all $0$.\nThe connection is torsion-free exactly when $\\Gamma^k_{ij} = \\Gamma^k_{ji}$ in every chart.\n",
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