High Quality Content by WIKIPEDIA articles! In differential geometry, a hyperkahler 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). Hyperkahler manifolds are special classes of Kahler manifolds. They can be thought of as quaternionic analogues of Kahler manifolds. All hyperkahler 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)). Hyperkahler manifolds were defined by E. Calabi in 1978.

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