Quaternion-Kähler manifold

Builds on Riemannian manifold Holonomy Almost complex manifold Levi-Civita connection Hyperkähler manifold Einstein manifold

Definition

Let $n \ge 2$ and let $(M, g)$ be a connected Riemannian manifold of dimension $4n$ . Let $Sp(n) \cdot Sp(1) = (Sp(n) \times Sp(1)) / \set{\pm 1}$ , acting on $\mathbb{H}^n = \R^{4n}$ by $v \mapsto a v \bar{b}$ for $a \in Sp(n)$ and unit quaternions $b \in Sp(1)$ . The manifold $M$ is quaternion-Kähler if its holonomy group is contained in $Sp(n) \cdot Sp(1) \subseteq SO(4n)$ .

Equivalently, there is a rank $3$ subbundle $Q \subseteq \End(TM)$ such that:

$Q$ is preserved by the Levi-Civita connection;
near each point, $Q$ has a local frame $I, J, K$ of almost complex structures that are orthogonal for $g$ and satisfy $IJ = K$ .

Remarks

Example: the quaternionic projective space $\mathbb{H}P^n$ with its standard symmetric metric, whose holonomy is all of $Sp(n) \cdot Sp(1)$ .

Non-example: no manifold of dimension $6$ is quaternion-Kähler, since the dimension must be a multiple of $4$ .

For $n \ge 2$ , a quaternion-Kähler manifold is Einstein. Hyperkähler manifolds satisfy the definition, with holonomy in $Sp(n) \subseteq Sp(n) \cdot Sp(1)$ ; many authors exclude them by asking for nonzero scalar curvature. In dimension $4$ the holonomy condition is empty, since $Sp(1) \cdot Sp(1) = SO(4)$ , so one defines quaternion-Kähler $4$ -manifolds instead as self-dual Einstein manifolds.

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Tag	`W6J4MQ6D5N`
URI	`m2://W6J4MQ6D5N`
Kind	definition
Writing	W0 stub
MSC	53C26, 53C29
Defines	Quaternion-Kähler manifold
Uses	Riemannian manifold, Holonomy, Almost complex manifold, Levi-Civita connection, Hyperkähler manifold, Einstein manifold
Changed	2 November 2026

Quaternion-Kähler manifold

Definition

Remarks

Reading path

Further reading

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