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  "body": "\n## Definition\n\nLet $M$ be a complex manifold, and let $J$ be the integrable [[almost complex structure|9SYWRGVTRS]] given by multiplication by $i$ in holomorphic charts.\nA *Hermitian metric* on $M$ is a [[Riemannian metric|Q61VFHCH2J]] $g$ on $M$ such that $g(JX, JY) = g(X, Y)$ for all tangent vectors $X, Y$ at the same point.\nA *Hermitian manifold* is a complex manifold with a Hermitian metric.\n\nThe *fundamental form* of $g$ is the [[differential $2$-form|0E3G0GRPZ6]] $\\omega(X, Y) = g(JX, Y)$.\nThe metric $g$ is *Kähler* if $d\\omega = 0$, where $d$ is the [[exterior derivative|004FRCXH6M]], and then $(M, g)$ is a *Kähler manifold*.\n\n## Remarks\n\nExample: $\\C^n = \\R^{2n}$ with the Euclidean metric $g_0$ and $J$ the multiplication by $i$.\nMultiplication by $i$ is an orthogonal map of $\\R^{2n}$, so $g_0(JX, JY) = g_0(X, Y)$.\nIts fundamental form is $\\omega = \\sum_k dx_k \\wedge dy_k$, which has constant coefficients, so $d\\omega = 0$ and $\\C^n$ is Kähler.\n\nNon-example: on $\\C = \\R^2$ the metric $g = dx^2 + 4\\, dy^2$ is Riemannian but not Hermitian.\nHere $J\\partial_x = \\partial_y$, so $g(J\\partial_x, J\\partial_x) = 4 \\ne 1 = g(\\partial_x, \\partial_x)$.\n\nThe form $h = g - i\\omega$ is a [[Hermitian form|48P4A1CX85]] on each tangent space viewed as a complex vector space.\nEvery complex manifold has Hermitian metrics: given any Riemannian metric $g$, the metric $g(X, Y) + g(JX, JY)$ is Hermitian.\nThe same definition with an almost complex structure that need not be integrable gives *almost Hermitian manifolds*; by the [[Newlander–Nirenberg theorem|VCT848WV1T]] the integrable case is the complex case.\nEvery complex submanifold of a Kähler manifold is Kähler with the induced metric.\n",
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