---
tag: NAJBH8GJMM
kind: concept
name: Quaternion-Kähler manifold
msc: [53C26, 53C29]
specializes: [7CBT4SM83Y]
sources:
  - url: https://en.wikipedia.org/wiki/Quaternion-K%C3%A4hler_manifold
same_as:
  wikidata: Q7269561
  wikipedia: Quaternion-Kähler manifold
---

## Summary

A quaternion-Kähler manifold is a Riemannian manifold of dimension $4n$ whose holonomy group lies in $Sp(n) \cdot Sp(1)$.
Locally it carries three almost complex structures that multiply like the quaternion units.
Parallel transport preserves the bundle they span, but not each of them.
For $n \ge 2$ such manifolds are Einstein, and despite the name they are in general not Kähler.
Quaternionic projective space $\mathbb{H}P^n$ is the basic example.

## Explanations

### L1 human

A [[hyperkähler manifold|87AWY7NQS0]] has three fixed ways to multiply by quaternion units.
In a quaternion-Kähler manifold the three ways may rotate among themselves as one moves around the manifold.
Only the three-dimensional family they span is well defined.
The quaternionic projective space, the quaternion analogue of the complex projective space, is the model example.