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
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If the almost complex structures are instead not assumed to be integrable, the manifold is called quaternionic, or almost hypercomplex.
Examples
Every hyperkähler manifold is also hypercomplex. The converse is not true. The Hopf surface
(with
In 1988, left-invariant hypercomplex structures on some compact Lie groups were constructed by the physicists Ph. Spindel, A. Sevrin, W. Troost, A. Van Proeyen. In 1992, D. Joyce rediscovered this construction, and gave a complete classification of left-invariant hypercomplex structures on compact Lie groups. Here is the complete list.
where
It is remarkable that any compact Lie group becomes hypercomplex after it is multiplied by a sufficiently big torus.
Basic properties
Hypercomplex manifolds as such were studied by Charles Boyer in 1988. He also proved that in real dimension 4, the only compact hypercomplex manifolds are the complex torus
Much earlier (in 1955) M. Obata studied affine connection associated with almost hypercomplex structures (under the former terminology of Charles Ehresmann of almost quaternionic structures). His construction leads to what Edmond Bonan called the Obata connection which is torsion free, if and only if, "two" of the almost complex structures
Twistor spaces
There is a 2-dimensional sphere of quaternions