Consider a smooth action $G\curvearrowright M$ of a Lie group on a manifold.
Suppose that an orbit $G\cdot p$ is closed. Is the orbit an embedded submanifold.
In general we know that the orbits are injective immersed submanifolds and if an orbit is embedded, then it is closed. Does the converse hold?
If the action is proper then the orbits are embedded, so the interesting question is about non proper actions.