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  "body": "\n## Summary\n\nThe curvature form of a [[connection on a principal bundle|NJ21PN5ZGJ]] is a $2$-form on the total space.\nIts values lie in the Lie algebra of the structure group.\nIt measures the failure of the horizontal distribution of the connection to be integrable, and it vanishes exactly when the connection is flat.\nFor connections on vector bundles it becomes a $2$-form with values in endomorphisms, and for the Levi-Civita connection it recovers the Riemann curvature tensor.\nIts invariant polynomials give the characteristic forms of Chern–Weil theory.\n\n## Explanations\n\n### L1 human\n\nA connection tells how to move horizontally from fibre to fibre.\nIf you move around a tiny loop in the base and lift the loop horizontally, you may come back to a different point of the same fibre.\nThe curvature form records this failure, loop direction by loop direction.\nIn electromagnetism, where the group is the circle, the curvature form is the electromagnetic field.\n",
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