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  "body": "\n## Statement\n\nLet $(M, g)$ be a [[pseudo-Riemannian manifold|GH6CXWFNDJ]].\nThen there is exactly one [[affine connection|JX65KSVHSJ]] $\\nabla$ on $M$ with the following two properties, for all smooth vector fields $X, Y, Z$ on $M$:\n\n1. $\\nabla$ is torsion-free: $\\nabla_X Y - \\nabla_Y X = [X, Y]$;\n2. $\\nabla$ is compatible with $g$: $X\\bigl(g(Y, Z)\\bigr) = g(\\nabla_X Y, Z) + g(Y, \\nabla_X Z)$.\n\nThis connection is the [[Levi-Civita connection|CAZYGE1T4E]] of $(M, g)$.\n\n## Remarks\n\nThe proof writes the compatibility identity three times with $X, Y, Z$ permuted cyclically and combines them using torsion-freeness.\nThis gives the Koszul formula\n$$2 g(\\nabla_X Y, Z) = X g(Y, Z) + Y g(X, Z) - Z g(X, Y) + g([X, Y], Z) - g([X, Z], Y) - g([Y, Z], X).$$\nSince $g$ is nondegenerate, the right side determines $\\nabla_X Y$, which gives uniqueness.\nFor existence, one checks that the vector field defined by the Koszul formula is an affine connection with both properties.\nIn coordinates the formula gives the [[Christoffel symbols|FZBA1HA9ZM]] $\\Gamma^k_{ij} = \\tfrac{1}{2} \\sum_l g^{kl}(\\partial_i g_{jl} + \\partial_j g_{il} - \\partial_l g_{ij})$.\nBoth conditions are needed for uniqueness: there are many metric-compatible connections with [[torsion|JZNQAN00ZJ]], and many torsion-free connections that do not preserve $g$.\n\n## Explanations\n\n### L1 human\n\nA metric measures lengths and angles at each point, but it does not by itself say how to differentiate vector fields.\nThis theorem says that once one asks the differentiation rule to keep lengths and angles and not to twist, there is no choice left: exactly one rule works.\nSo every Riemannian manifold comes with its own canonical way of differentiating.\n\n### L2 human\n\nThe theorem is why geodesics, parallel transport and curvature are invariants of the metric alone.\nThe Koszul formula is also a practical tool: it computes the connection of a left-invariant metric on a Lie group or of a warped product directly from brackets.\nThe same proof works for any nondegenerate metric, so the result holds for Lorentzian manifolds as used in general relativity.\nFor a surface in $\\R^3$, the Levi-Civita connection is the tangential part of the ordinary derivative.\nThis is one way to see that Gaussian curvature depends only on the first fundamental form.\n",
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