Quaternion-Kähler manifold

In differential geometry, a quaternion-Kähler manifold (or quaternionic Kähler manifold) is a Riemannian 4n-manifold whose Riemannian holonomy group is a subgroup of Sp(n)·Sp(1) for some ${\displaystyle n\geq 2}$. Here Sp(n) is the sub-group of ${\displaystyle SO(4n)}$ consisting of those orthogonal transformations that arise by left-multiplication by some quaternionic ${\displaystyle n\times n}$ matrix, while the group ${\displaystyle Sp(1)=S^{3}}$ of unit-length quaternions instead acts on quaternionic ${\displaystyle n}$-space ${\displaystyle {\mathbb {H} }^{n}={\mathbb {R} }^{4n}}$ by right scalar multiplication. The Lie group ${\displaystyle Sp(n)\cdot Sp(1)\subset SO(4n)}$ generated by combining these actions is then abstractly isomorphic to ${\displaystyle [Sp(n)\times Sp(1)]/{\mathbb {Z} }_{2}}$.

Although the above loose version of the definition includes hyperkähler manifolds, the standard convention of excluding these will be followed by also requiring that the scalar curvature be non-zero— as is automatically true if the holonomy group equals the entire group Sp(n)·Sp(1).