I’ve heard of the Stiefel manifold but never worked with it directly but I think I’ll start to.
I like that the (n)-dimensional sphere is just a special case of the Steifel manifold.[S^n=SO(n+1)/SO(n)=V_1(\mathbb{R}^{n+1})]The proof of the theorem which states[H^\bullet_G(M)\simeq H^\bullet_{dR}(M)]carries over virtually unchanged from the Lie group case if we assume that (G) acts transitively on (M). In other words, (M = G/H). If the (G)-action is not transitive, I’m not sure if the theorem is still true. In any case, the theorem remains true for the Steifel manifold so no problem with using the theorem to calculate the de Rham cohomology.
If you happen to find out if the theorem is still true for all actions (not just transitive ones), please let me know. I think it should remain true for all actions but the proof becomes more complicated.
Update: I’m pretty sure the result is true for all (G)-actions, not just transitive ones. The proof just requires a little modification. It’s not as difficult as I thought.