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In mathematics, a hyperkähler manifold is a Riemannian manifold of dimension 4k and holonomy group Sp(k). 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.
Every hyperkähler manifold M has a 2-sphere of complex structures with respect to which the metric is Kähler. In particular, there are three distinct complex structures,
- <math>J_1, J_2<math> and <math>J_3<math>
such that
- <math>J_1J_2 = J_3<math>.
Any linear combination
- <math>a_1J_1 + a_2J_2 + a_3J_3<math>
with
- <math>a_1^2 + a_2^2 + a_3^2 = 1<math>
is also a complex structure on M.
Compact hyperkähler 4-manifolds are called K3 surfaces. These have been extensively studied using techniques from algebraic geometry. Noncompact hyperkähler 4-manifolds which are asymptotic to H/G, where H denotes the quaternions and G is a finite subgroup of Sp(1), are known as Asymptotically locally Euclidean, or ALE, spaces. These space are studied in physics under the name gravitational instantons.
See also: Calabi-Yau manifold