Quaternion-Kähler symmetric space
- Quaternion-Kähler symmetric space
In differential geometry, a quaternion-Kähler symmetric space or Wolf space is a quaternion-Kähler manifold which, as a Riemannian manifold, is a Riemannian symmetric space. Any quaternion-Kähler symmetric space with positive Ricci curvature is compact and simply connected, and is a Riemannian product of quaternion-Kähler symmetric spaces associated to compact simple Lie groups.
For any compact simple Lie group "G", there is a unique "G"/"H" obtained as a quotient of "G" by a subgroup
:
Here, Sp(1) is the compact form of the SL(2)-triple associated with the highest root of "G", and "K" its centralizer in "G". These are classified as follows.
The twistor spaces of quaternion-Kähler symmetric spaces are the homogeneous holomorphic contact manifolds, classified by Boothby: they are the adjoint varieties of the complex semisimple Lie groups.
References
* Besse, Arthur Lancelot, "Einstein Manifolds", Springer-Verlag, New York (1987)
* Salamon, Simon, "Quaternionic Kähler manifolds", Invent. Math. 67 (1982), 143–171.
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