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. The Lie group 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 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).
Early history Marcel Berger's 1955 paper on the classification of Riemannian holonomy groups first raised the issue of the existence of non-symmetric manifolds with holonomy Sp(n)·Sp(1). Interesting results were proved in the mid-1960s in pioneering work by Edmond Bonan and Kraines who have independently proven that any such manifold admits a parallel 4-form Ω {\displaystyle \Omega } . A quaternion-Kähler analog of the hard Lefschetz theorem was later proved by Bonan. In the context of Berger's classification of Riemannian holonomies, quaternion-Kähler manifolds constitute the only class of irreducible, non-symmetric manifolds of special holonomy that are automatically Einstein, but not automatically Ricci-flat. If the Einstein constant of a simply connected manifold with holonomy in S p ( n ) S p ( 1 ) {\displaystyle Sp(n)Sp(1)} is zero, where n ≥ 2 {\displaystyle n\geq 2} , then the holonomy is actually contained in S p ( n ) {\displaystyle Sp(n)} , and the manifold is hyperkähler. This case is excluded from the definition by declaring quaternion-Kähler to mean not only that the holonomy group is contained in S p ( n ) S p ( 1 ) {\displaystyle Sp(n)Sp(1)} , but also that the manifold has non-zero (constant) scalar curvature. With this convention, quaternion-Kähler manifolds can thus be naturally divided into those for which the Ricci curvature is positive, and those for which it is instead negative.
Examples There are no known examples of compact quaternion-Kähler manifolds that are not locally symmetric. (Again, hyperkähler manifolds are excluded from the discussion by fiat.) On the other hand, there are many symmetric quaternion-Kähler manifolds; these were first classified by Joseph A. Wolf, and so are known as Wolf spaces. For any simple Lie group G, there is a unique Wolf space G/K obtained as a quotient of G by a subgroup K = K 0 ⋅ SU ( 2 ) {\displaystyle K=K_{0}\cdot \operatorname {SU} (2)} , where
S U ( 2 ) {\displaystyle SU(2)} is the subgroup associated with the highest root of G, and K0 is its centralizer in G. The Wolf spaces with positive Ricci curvature are compact and simply connected. For example, if
G = S p ( n + 1 ) {\displaystyle G=Sp(n+1)} , the corresponding Wolf space is the quaternionic projective space
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