Example of a homogeneous space that is not linear group orbit Give an example of a manifold $ M $ which is homogeneous (isomorphic to the quotient of a Lie group by a closed subgroup) but which is not a linear orbit. In other words, $ M $ is homogeneous but for any Lie group $ G $ any representation $ \pi: G \to GL(V) $ and any $ v \in V $ then the orbit
$$
\mathcal{O}_v:=\{ \pi(g)v:g \in G\} 
$$
is not diffeomorphic to $ M $.

For compact manifolds being a linear group orbit is equivalent to being Riemannian homogeneous see
https://mathoverflow.net/questions/409511/compact-linear-group-orbit-equivalent-to-linear-compact-group-orbit
so a compact linear group orbit is aspherical if and only if it is a torus. See
Riemannian homogeneous aspherical iff flat torus
Every compact solvmanifold is aspherical. So every compact solvmanifold that is not a torus is an example of a homogeneous manifold that is not a linear group orbit. Example include the Klein bottle in 2d and many manifolds with flat, Nil or Solv geometry in 3d see for example
https://mathoverflow.net/questions/414531/3-dimensional-solvmanifolds-and-thurston-geometries
