Inferring rank of a map from the rank of its quotient with knowledge of dimension of space modded out by?

Question

Suppose that $\mu: G \times M \rightarrow M$ is a smooth (we can add "proper" if we like) action of a Lie group on a smooth manifold. Fix a point $p \in M$. We can produce the "orbit map" for $p$ by defining $\mu_p:G \rightarrow M$ by $g \mapsto gp$.

It is known that if $G_p$ is the isotropy subgroup of $p$ in $G$, that $\mu_p$ passes to an injective immersion $\tilde{\mu_p}:\frac{G}{G_p} \rightarrow M$ whose image is the orbit $G(p)$ (in the "proper" case, this is a smooth embedding, and the orbits are embedded submanifolds).

I want to show that $d(\mu_p)_e$ ($\mu_p$, not the quotient map) is surjective onto $T_pG(p)$. Let's say we know the dimension of $G$, and of $G_p$, so we know the dimension of the orbit. Let's call these things $\dim G$ and $\dim G_p$, from which $\dim \frac{G}{G_p}$ follows. $\tilde{\mu_p}$ is then a map of constant rank $\dim \frac{G}{G_p}$.

The question is: can we infer anything about the rank of $\mu_p$ from this information? If not, is there something we could add, or is there some other way to conclude $d(\mu_p)_e$ is a surjection?

Answer 1

Here are some ways I can think of.

1. Assume that $\mu_p:G\to M$ (whose image is $G\cdot p$) is such that the target-restricted map $m_p:G\to G\cdot p$ is continuous (in general since $G\cdot p$ is only an immersed submanifold of $M$, its topology isn’t necessarily the induced subset topology, so this continuity is not automatic). Then, from the local canonical form for injective immersions $\iota_p:G\cdot p\to G$, smoothness of $\mu_p$ implies that of $m_p$. So, $m_p:G\to G\cdot p$ is a smooth, surjective map with the same constant rank as $\mu_p$. Hence, by the global rank theorem (see Lee’s smooth manifolds) it follows that $m_p$ is a submersion (this step uses Baire’s category theorem), so the constant rank of these maps is $\dim(G\cdot p)$.
2. Alternatively, if you assume that $G\cdot p$ is locally closed in $M$, then you can show by a Baire-category argument that it must be an embedded submanifold. In this case, the mapping $m_p$ defined above is automatically closed (because embedded submanifolds have the subspace topology so restricting target spaces of continuous maps preserves continuity). Now, run the argument as before: $m_p$ is a surjective, constant rank map, hence by the global rank theorem, must be a submersion.
3. Finally, you could simply assume at the outset that the action is nice enough that $\pi:M\to M/G$ is a smooth submersion (the latter having the quotient topology). In this case, every orbit is an embedded submanifold (regular-value theorem), so we can use the argument in (2).

Without one of these types of “mild” extra assumptions, I’m not sure how to argue it.