Let $M = G/K$ be a compact homogeneous space, i.e. $G$ a connected compact Lie group, and $K$ a closed subgroup inside $G$ that contains no nontrivial normal subgroup of $G$.

1. If $r(G) > r(K)$ (here, $r$ means the rank of maximal torus), is it true that $G/K$ admits free $S^1$-action? If so, why? If not, what is a counterexample? (Note: In “Homogeneous Spaces with Non-Vanishing Euler Characteristics” by HC Wang (1949), it was proven that $\chi(G/K) \ne 0 \Leftrightarrow r(G) = r(K)$.)
2. If 1. is unknown, then, restricting to case when $G$ semisimple, is 1. known?

Any input would be really helpful. Thanks a lot!

Let $M = G/K$ be a compact homogeneous space, i.e. $G$ a connected compact Lie group, and $K$ a closed subgroup inside $G$ that contains no nontrivial normal subgroup of $G$.

1. If $r(G) > r(K)$ (here, $r$ means the rank of maximal torus), is it true that $G/K$ admits free $S^1$-action? If so, why? If not, what is a counterexample?
2. What if we further assume $K$ is connected? (Note: In “Homogeneous Spaces with Non-Vanishing Euler Characteristics” by HC Wang (1949), it was proven that $\chi(G/K) \ne 0 \Leftrightarrow r(G) = r(K)$.)
3. If 1. is unknown, then, restricting to case when $G$ semisimple, is 1. known?

Any input would be really helpful. Thanks a lot!