Spin structures on compact homogeneous pseudo-Riemannian manifolds

Abstract

We study spin structures on compact simply-connected homogeneous pseudo-Riemannian manifolds (M = G/H, g) of a compact semisimple Lie group G. We classify flag manifolds F = G/H of a compact simple Lie group which are spin. This yields also the classification of all flag manifolds carrying an invariant metaplectic structure. Then we investigate spin structures on principal torus bundles over flag manifolds, i.e. C-spaces, or equivalently simply-connected homogeneous complex manifolds M=G/L of a compact semisimple Lie group G. We study the topology of M and we provide a sufficient and necessary condition for the existence of an (invariant) spin structure, in terms of the Koszul form of F. We also classify all C-spaces which are fibered over an exceptional spin flag manifold and hence they are spin.

References in corpus (1)

• The Srni lectures on non-integrable geometries with torsion

Cited by in corpus (6)

• Dirac operators for matrix algebras converging to coadjoint orbits
• Invariant connections and $\nabla$-Einstein structures on isotropy irreducible spaces
• Parallel spinors on Lorentzian Weyl spaces
• Homogeneous 8-manifolds admitting invariant Spin(7)-structures
• Cohomogeneity one Kaehler and Kaehler-Einstein manifolds with one singular orbit, II
• Homogeneous Einstein metrics on non-Kähler C-spaces