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.

Cited by in corpus (5)

• Connections on non-symmetric (generalized) Riemannian manifold and gravity
• Non-symmetric Riemannian gravity and Sasaki-Einstein 5-manifolds
• Instantons in six dimensions and twistors
• Instantons on the six-sphere and twistors
• Totally non cohomologous to zero fibrations and amenable Clifford-Klein forms of 3-symmetric spaces