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  "body": "\n## Definition\n\nLet $(M, g)$ be a [[Riemannian manifold|7CBT4SM83Y]] with [[Levi-Civita connection|CAZYGE1T4E]] $\\nabla$.\nThe manifold $M$ is *hyperkähler* if it carries three [[almost complex structures|9SYWRGVTRS]] $I$, $J$, $K$ such that:\n\n1. $g(IX, IY) = g(JX, JY) = g(KX, KY) = g(X, Y)$ for all tangent vectors $X, Y$;\n2. $\\nabla I = \\nabla J = \\nabla K = 0$;\n3. $IJ = K$.\n\n## Remarks\n\nFrom $I^2 = J^2 = -\\id$ and $IJ = K$ with $K^2 = -\\id$ one gets the quaternion relations $IJ = -JI = K$.\nClauses 1 and 2 say that $(M, g)$ is [[Kähler|CK1DCPC3D3]] for each of $I$, $J$, $K$, and in fact for $aI + bJ + cK$ whenever $a^2 + b^2 + c^2 = 1$.\nEquivalently, the [[holonomy|WHDZ3MK56B]] group of $g$ is contained in $Sp(n)$, where $\\dim M = 4n$.\n\nExample: $\\mathbb{H}^n = \\R^{4n}$ with the flat metric and $I, J, K$ given by left multiplication by the quaternions $i, j, k$ on each coordinate.\nThese maps are orthogonal, have constant coefficients so they are parallel, and satisfy $ij = k$.\n\nNon-example: $\\C P^2$ with the Fubini–Study metric is Kähler of real dimension $4$, but it is not hyperkähler.\nA hyperkähler metric is Ricci-flat, while the Fubini–Study metric has positive Ricci curvature.\n",
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