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  "body": "\n## Summary\n\nThe Levi-Civita connection of a [[pseudo-Riemannian manifold|GH6CXWFNDJ]] is the [[affine connection|JX65KSVHSJ]] that has no [[torsion|JZNQAN00ZJ]] and preserves the metric.\nBy the [[fundamental theorem of Riemannian geometry|78JRW8KTH8]] there is exactly one such connection.\nIt is the connection used to define geodesics, parallel transport and curvature of a Riemannian manifold.\n\n## Explanations\n\n### L1 human\n\nOn a curved surface there is no obvious way to compare vectors at different points, so differentiating a vector field is not obvious.\nA connection is a rule that does this, and there are many such rules.\nThe Levi-Civita connection is the one rule that respects lengths and angles and does not twist.\nOn a surface in $\\R^3$ it is what one gets by differentiating in $\\R^3$ and then projecting onto the tangent plane.\nFor $\\R^n$ with the usual metric it is ordinary differentiation of the components.\n",
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