Question on group actions on vector spaces

[mnb96]

Hello,
If I am given a vector space (e.g. $\mathbb{R}^n$), and a group $G$ that acts on $\mathbb{R}^n$, what are the conditions that $G$ must satisfy so that for any given $x\in\mathbb{R}^n$ its orbit $Gx$ is a manifold ?

 

[eok20]

If G is a Lie group acting smoothly on R^n, then the orbits will be immersed submanifolds.

 

[Office_Shredder]

The orbit has the same group structure as G does by multiplication $$gx \cdot hx = (gh)x$$ and the obvious isomorphism $$g \mapsto gx$$. So this means that your group G has to be a manifold. I'm guessing you'll want the multiplication to be continuous under the topology of the manifold or something, I'll have to think about itEDIT: Woops, not an isomorphism,. That's what happens when you use the word obvious.

Ok after thinking about this for a bit, I have to ask: did you perhaps want G to act linearly on the vector space?

 

[mnb96]

Ok after thinking about this for a bit, I have to ask: did you perhaps want G to act linearly on the vector space?

Uhm...Sorry, could you please specify what does "to act linearly" mean?
Thanks

 

[Office_Shredder]

Every element of G is a linear map, so along with the normal group action rules you have

$$g \cdot (\alpha x + \beta y) = \alpha g \cdot x + \beta g \cdot y$$

Otherwise the fact that we're living in a vector space is a happy coincidence and has very little bearing on the problem

 

[mnb96]

Ok. I got it, so yes...the group has to act linearly.