Hyperkähler manifold

In differential geometry, a hyperkähler manifold is a Riemannian manifold of dimension 4 k {displaystyle 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 {displaystyle k} -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(2k)). In differential geometry, a hyperkähler manifold is a Riemannian manifold of dimension 4 k {displaystyle 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 {displaystyle k} -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(2k)). 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