Compact homogeneous manifolds with integrable invariant distributions, and scalar curvature

It is proved that for a compact simply-connected effective homogeneous space $G/H$ of a connected compact Lie group $G$ by a closed subgroup $H$ the following conditions are equivalent:

• (1) Every $G$-invariant distribution on $G/H$ is integrable.
• (2) The space $G/H$ is of normal type in the sense of Bergery.
• (3) Every $G$-invariant Riemannian metric on $G/H$ has positive scalar curvature.
• (4) The space $G/H$ is isomorphic to a direct product of compact simplyconnected strongly isotropy-irreducible homogeneous spaces.