Lie group actions and submanifolds

In summary: However, for the purposes of this question, it's sufficient to show that if the group G is acting smoothly on a smooth manifold, then the orbit space of the points under G is an immersed submanifold. Thanks for your help!
  • #1
mnb96
715
5
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.
 
Physics news on Phys.org
  • #2
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.
 
  • #3
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.
 
  • #4
Fix a p in R^2 and consider the map G -> R^2 sending g to gp.
 
  • #5
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?
 
  • #6
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).
 
  • #7
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?
 
  • #8
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! :smile:).
 

1. What is a Lie group action?

A Lie group action is a mathematical concept that describes how a group (a set of elements with a binary operation) acts on a manifold (a space that locally resembles Euclidean space). This action preserves the structure of the manifold, such as its distances and angles, and can be used to study symmetries and transformations.

2. How are Lie group actions related to submanifolds?

Lie group actions can be used to study the geometry and topology of submanifolds, which are lower-dimensional manifolds embedded within a higher-dimensional manifold. The action of a Lie group on a submanifold can reveal its symmetries and help classify and understand its properties.

3. What is the significance of Lie group actions and submanifolds in physics?

Lie group actions and submanifolds have many applications in physics, particularly in the study of symmetries and conservation laws. For example, in the theory of relativity, the group of Lorentz transformations acts on spacetime, revealing its symmetries and allowing for the study of physical laws that are invariant under these transformations.

4. How are Lie group actions and submanifolds used in differential geometry?

In differential geometry, Lie group actions are often used to define and study geometric structures such as connections and curvature. Submanifolds provide a way to study these structures on a local level, and the action of a Lie group on a submanifold can reveal its geometric properties.

5. Can Lie group actions and submanifolds be used in practical applications?

Yes, Lie group actions and submanifolds have practical applications in fields such as robotics, computer graphics, and computer vision. They can be used to model and analyze the motion of robotic arms and objects in 3D space, create realistic animations, and develop algorithms for object recognition and tracking.

Similar threads

  • Linear and Abstract Algebra
Replies
1
Views
1K
  • Linear and Abstract Algebra
Replies
3
Views
687
  • Linear and Abstract Algebra
Replies
10
Views
1K
  • Linear and Abstract Algebra
Replies
15
Views
2K
  • Linear and Abstract Algebra
Replies
1
Views
1K
  • Linear and Abstract Algebra
Replies
4
Views
1K
Replies
0
Views
287
  • Advanced Physics Homework Help
Replies
14
Views
1K
  • Differential Geometry
Replies
9
Views
419
  • Linear and Abstract Algebra
Replies
3
Views
2K
Back
Top