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 n ≥ 2 {\displaystyle n\geq 2} . Here Sp(n) is the sub-group of S O ( 4 n ) {\displaystyle SO(4n)} consisting of those orthogonal transformations that arise by left-multiplication by some quaternionic n × n {\displaystyle n\times n} matrix, while the group S p ( 1 ) = S 3 {\displaystyle Sp(1)=S^{3}} of unit-length quaternions instead acts on quaternionic n {\displaystyle n} -space H n = R 4 n {\displaystyle {\mathbb {H} }^{n}={\mathbb {R} }^{4n}} by right scalar multiplication.

Source: Wikipedia — Quaternion-Kähler manifold (CC BY-SA 4.0)

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 n ≥ 2 {\displaystyle n\geq 2} . Here Sp(n) is the sub-group of S O ( 4 n ) {\displaystyle SO(4n)} consisting of those orthogonal transformations that arise by left-multiplication by some quaternionic n × n {\displaystyle n\times n} matrix, while the group S p ( 1 ) = S 3 {\displaystyle Sp(1)=S^{3}} of unit-length quaternions instead acts on quaternionic n {\displaystyle n} -space H n = R 4 n {\displaystyle {\mathbb {H} }^{n}={\mathbb {R} }^{4n}} by right scalar multiplication.

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Source: Wikipedia "Quaternion-Kähler manifold" · CC BY-SA 4.0

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