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High Quality Content by WIKIPEDIA articles! 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 an n-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.