A classification of integrable twistor spaces of various kinds over Riemannian symmetric spaces. Given an even dimensional Riemannian manifold N, the bundle J(N) over N of orthogonal complex structures on the tangent spaces of N has a natural almost complex structure and complex horizonal plane field. Unless N has constant sectional curvature, however, the almost complex structure on J(N) will not be integrable. A twistor subspace Z of J(N) is a sub-bundle that is an almost complex submanifold of J(N) and that has the additional properties that the induced almost complex structure on Z is actually integrable and that the horizontal plane field is tangent to Z.
In this paper, I find all of the twistor subspaces of J(N) when N is a Riemannian symmetric space. These all turn out to be orbits of the isometry group of N and so can be classified by examining the root systems of the simple Lie groups.
In the last section, I construct a different sort of twistor space over each Riemannian symmetric space.
For further developments in this area, consult the works of F. Burstall, J. Rawnsley, S. Salamon, and J. Wood.