Lie group actions and submanifolds

[mnb96]

Hello,

Let's suppose that I have a Lie group G parametrized by one real scalar t and acting on ℝ².
Is it generally correct to say that the orbits of the points of ℝ² under the group action are one-dimensional submanifolds of ℝ², because G is parametrized by one single scalar?

If so, how can I prove this statement?

Thanks.

 

[morphism]

What if the action has a fixed point at p? Will the orbit through p be one-dimensional?

If your G is acting smoothly on R^2, you can at least say that the orbit through each point is an immersed submanifold of R^2; it will generally be either 1- or 0-dimensional.

 

[mnb96]

thanks a lot!

you are right. I am just thinking of the action of the rotation group SO(2) on ℝ²; clearly the point at (0,0) will remain unchanged, hence its orbit is 0-dimensional.

Do you have any hint to suggest in order to prove these facts? I mean, proving that the orbits are immersed submanifolds of R^2.

 

[morphism]

Fix a p in R^2 and consider the map G -> R^2 sending g to gp.

 

[mnb96]

I see...
I guess all I have to do is to prove that by letting G act on ℝ², the space ℝ² will be partitioned into equivalence classes (=the orbits), and that follows from the very fact that G is a group...maybe?

 

[morphism]

That's kind of besides the point. The image of the map I wrote down is precisely the orbit through p. So all that remains is to show that the map is an immersion - this will prove that the orbit is an immersed submanifold (by definition).

Be careful to note that while each orbit itself is an immersed submanifold, the orbit space R^2/G with the quotient topology need not even be Hausdorff (let alone a manifold).

 

[mnb96]

ok...
everything is almost clear. The only piece I am missing is how to prove that a mapping is an immersion (I am not familiar with this definition). Am I supposed to consider the mapping from the smooth manifold G (the Lie group parametrized by t) to the orbit of a point in R^2, take their derivatives with respect to t, and show that the map is injective?

 

[morphism]

I lied earlier - the map G -> R^2 isn't necessarily an immersion (e.g. if p is a fixed point). What we should be looking at is the induced map G/G_p -> R^2, where G_p = {g in G | gp=p} is the isotropy subgroup at p.

To rigorously prove that this map is an immersion you need to know a thing or two about differential geometry (in particular you need to know what "immersion" means! ).