Twistors and 3-symmetric spaces

Abstract

We describe complex twistor spaces over inner 3-symmetric spaces $G/H$, such that $H$ acts transitively on the fibre. Like in the symmetric case, these are flag manifolds $G/K$ where $K$ is the centralizer of a torus in $G$. Moreover, they carry an almost complex structure defined using the horizontal distribution of the normal connection on $G/H$, that coincides with the complex structure associated to a parabolic subgroup $P \subset G^{\mathbb C}$ if it is integrable. Conversely, starting from a complex flag manifold $G^{\mathbb C}/P$, there exists a natural fibration with complex fibres on a 3-symmetric space, called fibration of degree 3.