High Quality Content by WIKIPEDIA articles! In differential geometry, a quaternion-Kahler manifold (or quaternionic Kahler manifold) is a Riemannian manifold whose Riemannian holonomy group is a subgroup of Sp(n)·Sp(1). Another, more explicit, definition, uses a 3-dimensional subbundle H of End(TM) of endomorphisms of the tangent bundle to a Riemannian M. For M to be quaternion-Kahler, H should be preserved by the Levi-Civita connection and pointwise isomorphic to the imaginary quaternions, in such a way that unit imaginary quaternions in H act on TM preserving the metric. Notice that this definition includes hyperkahler manifolds. However, these are often excluded from the definition of a quaternion-Kahler manifold by imposing the condition that the scalar curvature is nonzero, or that the holonomy group is equal to Sp(n)·Sp(1).

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