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  "body": "\n## Definition\n\nLet $(M, g)$ be a [[pseudo-Riemannian manifold|GH6CXWFNDJ]], and write $\\mathfrak{X}(M)$ for the space of smooth vector fields on $M$.\nA *Levi-Civita connection* of $(M, g)$ is an [[affine connection|JX65KSVHSJ]] $\\nabla$ on $M$ such that:\n\n1. $\\nabla$ is torsion-free: $\\nabla_X Y - \\nabla_Y X = [X, Y]$ for all $X, Y \\in \\mathfrak{X}(M)$, where $[X, Y]$ is the [[Lie bracket|68FGQEBD6J]];\n2. $\\nabla$ is compatible with $g$: $X\\bigl(g(Y, Z)\\bigr) = g(\\nabla_X Y, Z) + g(Y, \\nabla_X Z)$ for all $X, Y, Z \\in \\mathfrak{X}(M)$.\n\n## Remarks\n\nExample: on $\\R^n$ with $g = \\sum_i dx_i^2$, let $\\nabla_X Y$ be the vector field whose components are $X(Y^1), \\ldots, X(Y^n)$.\nThen $\\nabla_X Y - \\nabla_Y X$ has components $X(Y^i) - Y(X^i)$, which is $[X, Y]$, and the product rule gives clause 2 for $g(Y, Z) = \\sum_i Y^i Z^i$.\n\nNon-example: on $\\R^3$ with the Euclidean metric, let $\\nabla_X Y = D_X Y + X \\times Y$, where $D$ is the flat connection above and $\\times$ is the cross product.\nThis is an affine connection, and clause 2 holds because $\\ip{X \\times Y}{Z} + \\ip{Y}{X \\times Z} = 0$.\nIts torsion is $\\nabla_X Y - \\nabla_Y X - [X, Y] = 2 \\, X \\times Y$, which is not zero, so clause 1 fails.\n\nThe two clauses determine $\\nabla$; this is the [[fundamental theorem of Riemannian geometry|78JRW8KTH8]].\nIn coordinates, $\\nabla_{\\partial_i} \\partial_j = \\sum_k \\Gamma^k_{ij} \\partial_k$ with the [[Christoffel symbols|FZBA1HA9ZM]]\n$$\\Gamma^k_{ij} = \\frac{1}{2} \\sum_l g^{kl} \\left( \\partial_i g_{jl} + \\partial_j g_{il} - \\partial_l g_{ij} \\right).$$\n",
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