Hypercomplex manifold

In differential geometry, a hypercomplex manifold is a manifold with the tangent bundle equipped with an action by the algebra of quaternions in such a way that the quaternions $I,J,K$ define integrable almost complex structures.

If the almost complex structures are instead not assumed to be integrable, the manifold is called quaternionic, or almost hypercomplex.^[1]

^ Manev, Mancho; Sekigawa, Kouei (2005). "Some Four-Dimensional Almost Hypercomplex Pseudo-Hermitian Manifolds". In S. Dimiev and K. Sekigawa (ed.). Contemporary Aspects of Complex Analysis, Differential Geometry and Mathematical Physics. Vol. 2005. Hackensack, NJ: World Sci. Publ. pp. 174–186. arXiv:0804.2814. doi:10.1142/9789812701763_0016. ISBN 978-981-256-390-3.