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  "body": "\n## Summary\n\nA Hermitian metric on a complex [[vector bundle|P4DZM975BV]] is a choice of [[Hermitian inner product|0NB0AWGJBW]] on each fibre that varies smoothly with the base point.\nIt lets one measure the length of sections and define the Chern connection and its curvature.\nEvery complex vector bundle on a paracompact manifold has one, by a partition of unity.\nA Hermitian metric on the holomorphic tangent bundle of a complex manifold is what makes it a [[Hermitian manifold|CK1DCPC3D3]].\n\n## Explanations\n\n### L1 human\n\nEach fibre of a complex vector bundle is a complex vector space, but it comes with no notion of length.\nA Hermitian metric gives each fibre an inner product, as on $\\C^r$, and asks that it change smoothly from fibre to fibre.\nThen each vector in the bundle has a length, and each section has a length function on the base.\nOn the trivial line bundle over $\\C$, the rule $\\abs{v}_z^2 = e^{-\\abs{z}^2} \\abs{v}^2$ is such a metric: it is the usual length, rescaled by a positive smooth factor at each point.\n",
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