---
tag: 87AWY7NQS0
kind: concept
name: Hyperkähler manifold
msc: [53C26, 53C29]
specializes: [CK1DCPC3D3]
sources:
  - url: https://en.wikipedia.org/wiki/Hyperk%C3%A4hler_manifold
same_as:
  wikidata: Q3082377
  wikipedia: Hyperkähler manifold
---

## Summary

A hyperkähler manifold is a Riemannian manifold that is Kähler with respect to three complex structures $I$, $J$, $K$ that multiply like the quaternion units.
Its dimension is a multiple of $4$, and its holonomy group lies in the compact symplectic group $Sp(n)$.
Hyperkähler manifolds are Ricci-flat, and they are one of the special holonomy cases in Berger's list.
Examples include K3 surfaces with their Ricci-flat metrics and many moduli spaces from gauge theory.

## Explanations

### L1 human

The quaternions extend the complex numbers with three square roots of $-1$, called $i$, $j$ and $k$, with $ij = k$.
A hyperkähler manifold lets one multiply tangent vectors by all three, in a way that respects lengths and stays the same when moved along curves.
The simplest example is the space $\mathbb{H}^n$ of quaternion vectors with its flat metric.
So each such manifold is a complex manifold in many ways at once, one for each unit imaginary quaternion.