Let $\phi:G\times M\rightarrow M$ be a smooth and transitive action of a real Lie group $G$ on a smooth real manifold $M$. Both $G$ and $M$ are assumed to be finite-dimensional but neither are assumed to be compact or connected.
For $g\in G$, the derivative $D\phi_g(x)$ of the partial map $\phi_g:M\rightarrow M, \phi_g(x):=\phi(g,x)$ at a point $x\in M$ is then a vector space isomorphism between the tangent spaces $T_x M$ and $T_{\phi_g(x)}M$.
The question is when does there exist a family $(|.|_x)_{x\in M}$ with each $|.|_x$ a norm on $T_x M$ such that $$\exists_{C>0}\ \forall_{g\in G,\ x\in M}\ \forall_{\eta\in T_{x}M}\quad |D\phi_g(x) [\eta]|_{\phi_g(x)} \leq C\cdot|\eta|_x\ ?$$
The special case $C=1$ is of course well studied (invariant norms on homogeneous spaces), so the answer is positive (for example) if the stabilizer of $x\in M$ is compact. This includes the cases where $G$ is compact, or acts properly. More generally, invariant norms exist on reductive homogeneous spaces; this includes the cases where the stabilizer is connected and semi-simple, or where the stabilizer is discrete. Invariant norms can also exist in the non-reductive case, for example if the stabilizer is normal in $G$.
But is the above condition (for arbitrary $C>0$) strictly weaker than the existence of an invariant norm or is it equivalent? Does such a family of norms maybe always exist? If not, what is a simple counter-example?
Note: Because $D\phi_{g^ {-1}}(\phi_g(x))$ is the inverse of $D\phi_g(x)$, such a $C$ is necessarily $\geq 1$.
