Hyperkähler

Hyperkähler is a term in differential geometry describing a Riemannian manifold that possesses a particularly rich structure. A Riemannian manifold is called hyperkähler if it admits a parallel G2 structure. This condition is equivalent to saying that the holonomy group of the manifold is a subgroup of G2.

A more concrete characterization of hyperkähler manifolds is that they admit a parallel complex structure J and a parallel 2-form Omega such that Omega(v, Jv) = 0 and Omega(Jv, Jw) = Omega(v, w) for all tangent vectors v, w. In essence, a hyperkähler manifold is a Kähler manifold whose Kähler form is parallel. This implies that the manifold is Ricci-flat, meaning its Ricci curvature tensor is identically zero.

The simplest examples of hyperkähler manifolds are the four-dimensional Euclidean space R^4 and the complex K3 surface. Higher-dimensional hyperkähler manifolds are also known to exist, and they play a significant role in various areas of theoretical physics, particularly in string theory and quantum field theory. The existence of a hyperkähler structure on a manifold often leads to powerful integrability conditions and special properties. The study of hyperkähler manifolds is an active area of research in mathematics.