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  "body": "\n## Definition\n\nLet $X$ be a smooth manifold and let $E \\to X$ be a smooth complex [[vector bundle|P4DZM975BV]].\nA *Hermitian metric* on $E$ is a family $h = (h_x)_{x \\in X}$ such that:\n\n1. For each $x \\in X$, $h_x \\colon E_x \\times E_x \\to \\C$ is a [[Hermitian inner product|0NB0AWGJBW]] on the fibre $E_x$.\n   So it is $\\C$-linear in the first argument, $h_x(w, v) = \\overline{h_x(v, w)}$, and $h_x(v, v) \u003e 0$ for $v \\ne 0$.\n2. For any two smooth local [[sections|WWX7H5W1RP]] $s, t$ of $E$ over an open set $U$, the function $x \\mapsto h_x(s(x), t(x))$ is smooth on $U$.\n\n## Remarks\n\nIn a local frame $e_1, \\dots, e_r$ of $E$ over $U$, a Hermitian metric is the same as a smooth map $H = (h(e_j, e_i))_{i,j}$ from $U$ to the positive definite Hermitian $r \\times r$ matrices.\nA vector bundle with a Hermitian metric is called a Hermitian vector bundle.\nWhen $X$ is a complex manifold and $E$ is its holomorphic tangent bundle, a Hermitian metric on $E$ is a Hermitian metric on $X$ in the sense of [[Hermitian manifolds|CK1DCPC3D3]].\n\nExample: on $\\mathcal{O}(-1) \\subset \\mathbb{P}^n \\times \\C^{n+1}$, restrict the standard inner product $\\sum_k v_k \\bar w_k$ of $\\C^{n+1}$ to each fibre.\nEach restriction is a Hermitian inner product, and for a local holomorphic section such as $s(z) = (1, z_1, \\dots, z_n)$ on $\\set{z_0 \\ne 0}$ one gets $h(s, s) = 1 + \\abs{z_1}^2 + \\cdots + \\abs{z_n}^2$, which is smooth.\n\nNon-example: on the trivial line bundle $\\C \\times \\C \\to \\C$, the rule $h_z(v, w) = \\abs{z}^2 v \\bar w$ is smooth, but $h_0(v, v) = 0$ for all $v$, so clause 1 fails at $z = 0$.\n",
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