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  "body": "\n## Summary\n\nA connection on a vector bundle is a rule for differentiating [[sections|WWX7H5W1RP]] of the bundle in the direction of tangent vectors.\nIt is needed because the fibres over different points are different vector spaces.\nSo there is no built-in way to compare a section at nearby points.\nA connection gives covariant derivatives, parallel transport, curvature and holonomy.\nConnections on the tangent bundle are the affine connections, such as the Levi-Civita connection.\nConnections on other bundles are the gauge fields of physics.\n\n## Explanations\n\n### L1 human\n\nA section of a vector bundle puts a vector in a separate vector space above each point.\nTo say whether such a section is \"constant\" you must compare vectors in different spaces, and nothing tells you how.\nA connection is the extra data that says how to differentiate the section, and so how to compare nearby fibres.\nOn the trivial bundle $\\R^n \\times \\R^r$ one can just differentiate each component, and this is the simplest connection.\nOn a twisted bundle, such as the tangent bundle of a sphere, there is no such obvious choice.\n",
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