Hyperkähler manifold

Builds on Riemannian manifold Almost complex manifold Levi-Civita connection Hermitian manifold Holonomy

Definition

Let $(M, g)$ be a Riemannian manifold with Levi-Civita connection $\nabla$ . The manifold $M$ is hyperkähler if it carries three almost complex structures $I$ , $J$ , $K$ such that:

$g(IX, IY) = g(JX, JY) = g(KX, KY) = g(X, Y)$ for all tangent vectors $X, Y$ ;
$\nabla I = \nabla J = \nabla K = 0$ ;
$IJ = K$ .

Remarks

From $I^2 = J^2 = -\id$ and $IJ = K$ with $K^2 = -\id$ one gets the quaternion relations $IJ = -JI = K$ . Clauses 1 and 2 say that $(M, g)$ is Kähler for each of $I$ , $J$ , $K$ , and in fact for $aI + bJ + cK$ whenever $a^2 + b^2 + c^2 = 1$ . Equivalently, the holonomy group of $g$ is contained in $Sp(n)$ , where $\dim M = 4n$ .

Example: $\mathbb{H}^n = \R^{4n}$ with the flat metric and $I, J, K$ given by left multiplication by the quaternions $i, j, k$ on each coordinate. These maps are orthogonal, have constant coefficients so they are parallel, and satisfy $ij = k$ .

Non-example: $\C P^2$ with the Fubini–Study metric is Kähler of real dimension $4$ , but it is not hyperkähler. A hyperkähler metric is Ricci-flat, while the Fubini–Study metric has positive Ricci curvature.

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Tag	`3ZCJRT8FPT`
URI	`m2://3ZCJRT8FPT`
Kind	definition
Writing	W0 stub
MSC	53C26, 53C29
Defines	Hyperkähler manifold
Uses	Riemannian manifold, Almost complex manifold, Levi-Civita connection, Hermitian manifold, Holonomy
Changed	2 November 2026

Hyperkähler manifold

Definition

Remarks

Reading path

Further reading

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