Hyperkähler manifold
In differential geometry, a hyperkähler manifold is a Riemannian manifold of dimension 4k and holonomy group contained in Sp(k) (here Sp(k) denotes a compact form of a symplectic group, identified with the group of quaternionic-linear unitary endomorphisms of a -dimensional quaternionic Hermitian space). Hyperkähler manifolds are special classes of Kähler manifolds. They can be thought of as quaternionic analogues of Kähler manifolds. All hyperkähler manifolds are Ricci-flat and are thus Calabi–Yau manifolds (this can be easily seen by noting that Sp(k) is a subgroup of SU(2k)).
Hyperkähler manifolds were defined by E. Calabi in 1978.
Every hyperkähler manifold M has a 2-sphere of complex structures (i.e. integrable almost complex structures) with respect to which the metric is Kähler.
In particular, it is an almost quaternionic manifold, meaning that there are three distinct complex structures, I, J, and K, which satisfy the quaternion relations
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