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In differential geometry, a quaternion-Kähler manifold is a Riemannian 4n-manifold whose Riemannian holonomy group is a subgroup of Sp·Sp for some n ≥ 2 {\displaystyle n\geq 2}. Here Sp is the sub-group of S O {\displaystyle SO} 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 = S 3 {\displaystyle Sp=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 Lie group S p ⋅ S p ⊂ S O {\displaystyle Sp\cdot Sp\subset SO} generated by combining these actions is then abstractly isomorphic to × S p ] / Z 2 {\displaystyle \times Sp]/{\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·Sp.