Compact homogeneous manifolds with integrable invariant distributions, and scalar curvature

Abstract

It is proved that for a compact simply-connected effective homogeneous space of a connected compact Lie group by a closed subgroup the following conditions are equivalent: (1) Every -invariant distribution on is integrable. (2) The space is of normal type in the sense of Bergery. (3) Every -invariant Riemannian metric on has positive scalar curvature. (4) The space is isomorphic to a direct product of compact simplyconnected strongly isotropy-irreducible homogeneous spaces.