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.