A hyper-Kähler manifold can be defined as a Riemannian manifold of dimension 4n with three covariantly constant orthogonal automorphisms I, J, K of the tangent bundle which satisfy the quaternionic identities I^2 = J^2 = K^2 = I J K = - 1, where -1 denotes the negative of the identity automorphism 1 = id on the tangent bundle. The term hyper-Kähler is sometimes written without a hyphen (as hyperKähler) or without capitalization (as hyperkähler). This definition is equivalent to several others commonly encountered in the literature; indeed, a manifold M^(4n) is said to be hyper-Kähler if and only if: 1.M is a holomorphically symplectic Kähler manifold with holonomy in Sp(n).

flat manifold | Kähler manifold | quaternion Kähler manifold | Riemannian manifold | symplectic manifold