Hyperkähler manifold

In differential geometry, a hyperkähler manifold is a Riemannian manifold ${\displaystyle (M,g)}$ endowed with three integrable almost complex structures ${\displaystyle I,J,K}$ that are Kähler with respect to the Riemannian metric ${\displaystyle g}$ and satisfy the quaternionic relations ${\displaystyle I^{2}=J^{2}=K^{2}=IJK=-1}$. In particular, it is a hypercomplex manifold. All hyperkähler manifolds are Ricci-flat and are thus Calabi–Yau manifolds.

Hyperkähler manifolds were first given this name by Eugenio Calabi in 1979.