Hyperkähler manifold  In differential geometry, a hyperkähler manifold is a Riemannian manifold of dimension formula_1 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 formula_2-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 the special unitary group SU(2"k")). Hyperkähler manifolds were defined by Eugenio 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 a hypercomplex manifold, meaning that there are three distinct complex structures, "I", "J," and "K," which satisfy the quaternion relations  Any linear combination  with formula_5 real numbers such that  is also a complex structure on "M". In particular, the tangent space "T""x""M" is a quaternionic vector space for each point "x" of "M". Sp("k") can be considered as the group of orthogonal transformations of formula_7 which are linear with respect to "I", "J" and "K". From this it follows that the holonomy of the manifold is contained in Sp("k"). Conversely, if the holonomy group of the Riemannian manifold "M" is contained in Sp("k"), choose complex structures "I""x", "Jx" and "K""x" on "T""x""M" which make "T""x""M" into a quaternionic vector space. Parallel transport of these complex structures gives the required quaternionic structure on "M". A hyperkähler manifold ("M","I","J","K"), considered as a complex manifold ("M","I"), is holomorphically symplectic (equipped with a holomorphic, non-degenerate 2-form). The converse is also true in the case of compact manifolds, due to Shing-Tung Yau's proof of the Calabi conjecture: Given a compact, Kähler, holomorphically symplectic manifold ("M","I"), it is always equipped with a compatible hyperkähler metric.