  in differential geometry, a hyperkÃ¤hler 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 k-dimensional quaternionic hermitian space). hyperkÃ¤hler manifolds are special classes of kÃ¤hler manifolds. they can be thought of as quaternionic analogues of kÃ¤hler manifolds. all hyperkÃ¤hler manifolds are ricci-flat and are thus calabiâyau manifolds (this can be easily seen by noting that sp(k) is a subgroup of su(2k)). hyperkÃ¤hler manifolds were defined by e. calabi in 1978. ==quaternionic structure== every hyperkÃ¤hler manifold m has a 2-sphere of complex structures (i.e. integrable almost complex structures) with respect to which the metric is kÃ¤hler. 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 :i^2 = j^2 = k^2 = ijk = -1.\, any linear combination :ai + bj + ck \, with a, b, c real numbers such that :a^2 + b^2 + c^2 = 1 \, 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 \mathbb{r}^{4k}=\mathbb{h}^k 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&nbsp;m. ==holomorphic symplectic form== a hyperkÃ¤hler 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 yau's proof of the calabi conjecture: given a compact, kÃ¤hler, holomorphically symplectic manifold (m,i), it is always equipped with a compatible hyperkÃ¤hler metric. such a metric is unique in a given kÃ¤hler class. compact hyperkÃ¤hler manifolds have been extensively studied using techniques from algebraic geometry, sometimes under a name holomorphically symplectic manifolds. due to bogomolov's decomposition theorem (1974), the holonomy group of a compact holomorphicall
Algebraic_geometry Complex_manifolds Riemannian_manifolds Structures_on_manifolds