Quaternion-Kähler manifold

In <a href="/facts/Differential_geometry/c04XmyLu">differential geometry</a>, a quaternion-Kähler manifold (or quaternionic Kähler manifold) is a Riemannian 4n-manifold whose <a href="/facts/Holonomy/GnxeTqI4">Riemannian holonomy group</a> 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. The <a href="/facts/Lie_group/a9jCQyjF">Lie group</a> 
 
 
 
 S
 p
 (
 n
 )
 ⋅
 S
 p
 (
 1
 )
 ⊂
 S
 O
 (
 4
 n
 )
 
 
 {\displaystyle Sp(n)\cdot Sp(1)\subset SO(4n)}
 
 generated by combining these actions is then abstractly isomorphic to 
 
 
 
 [
 S
 p
 (
 n
 )
 ×
 S
 p
 (
 1
 )
 ]
 
 /
 
 
 
 
 Z
 
 
 
 2
 
 
 
 
 {\displaystyle [Sp(n)\times Sp(1)]/{\mathbb {Z} }_{2}}
 
.
Although the above loose version of the definition includes <a href="/facts/Hyperk%C3%A4hler_manifold/PdQP4Kqm">hyperkähler manifolds</a>, the standard convention of excluding these will be followed by also requiring that the <a href="/facts/Scalar_curvature/tfgm3AuK">scalar curvature</a> be non-zero— as is automatically true if the holonomy group equals the entire group Sp(n)·Sp(1).

Quaternion-Kähler manifold open-in-new

Quaternion-Kähler manifold