I have a homogeneous Riemannian manifold X with isometry group Iso. Is Iso a maximal group? By maximal group, I mean that there does not exist another group G such that:
- Iso is a proper subgroup of G,
- G acts transitively on X by diffeomorphisms, and
- G has compact stabilizers Gx.
I know that if such a group exists, then there is a G-invariant metric on X. So, in other words, the question is: if I have a Riemannian metric d on X with isometry group Iso which is homogeneous, can there exist another Riemannian metric d' on X with isometry group Iso', such that Iso' contains properly the group Iso?