Proving Correspondence between SO(3)/SO(2) and S^2

  • Thread starter Thread starter quantum_smile
  • Start date Start date
  • Tags Tags
    Lie groups
AI Thread Summary
The discussion revolves around establishing a one-to-one correspondence between the coset space SO(3)/SO(2) and the two-dimensional sphere S^2. Participants note that SO(2) is represented within SO(3) by matrices of a specific form, and they explore the geometric and algebraic properties of these groups. A key approach involves defining a mapping from SO(3)/SO(2) to S^2, specifically using the transformation that maps cosets to points on the sphere. The challenge lies in demonstrating that this mapping is well-defined, one-to-one, and onto, emphasizing the geometric nature of the problem. Understanding the relationship between these mathematical structures is crucial for solving the homework problem.
quantum_smile
Messages
21
Reaction score
1

Homework Statement


Take the subgroup isomorphic to SO(2) in the group SO(3) to be the group of matrices of the form
<br /> \begin{pmatrix} g &amp; &amp; 0 \\ &amp; &amp; 0 \\ 0 &amp; 0 &amp; 1 \end{pmatrix}, g\in{}SO(2).<br />
Show that there is a one-to-one correspondence between the coset space of SO(3) by this subgroup and the two-dimensional sphere
<br /> SO(3)/SO(2)=S^2.<br />

Homework Equations


<br /> SO(3)/SO(2)=\{[A] | A\in{}SO(3)\}\\<br /> [A]:=\{B\in{}SO(3) | B = AH, H\in{}SO(2)\}<br />

The Attempt at a Solution


As a first step, I tried to get a more intuitive grasp of what SO(3)/SO(2) "is." However, all I've been able to get is a series of cumbersome matrix component equations, and I have yet to figure out how to really utilize the fact that all the matrices are orthogonal and that their determinants are 1.

Also, I've only learned very basic topology, so I don't know how to set up a one-to-one correspondence between SO(3)/SO(2) and S^2.

Any help/hints are very appreciated!
 
  • Like
Likes jakob1111
Physics news on Phys.org
If you have never worked with this kind of stuff, I don't this is at all obvious. What text, if any, doe your course use?

Let's start with ...

What is ##S^2##?
 
I'm using Rubakov - Classical Theory of Gauge Fields.

##S^2## is the 2-sphere in three-dimensional space. I was thinking - **if** I could prove that each element of SO(3)/SO(2) can be fully characterized by three real parameters such that their moduli sum to 1, then I could set up a one-to-one correspondence between each element of SO(3)/SO(2) and a set of Cartesian coordinates for S^2. I can't figure out how to do that though.
 
Well, I think that we should be more concrete as to what we are going to use as ##S^2##. For this example, ##S^2## is the subset of ##\mathbb{R}^3## given by

$$S^2 = \left\{ \mathbf{v} \in \mathbb{R}^3 | \mathbf{v} \cdot \mathbf{v} = 1 \right\}$$
 
Yes, it's the problem#6 of his book. The problem here lies in the fact that ##S^2## is a manifold, while ##SO(3)/SO(2)## is a Lie group (thus also a manifold). So this problem is also geometric, not only algebraic.

So I think problems 6, 7 and 8 are all linked together. If you read about homogenous spaces and group actions on ##\mathbb{R}^3##, then you should be able to solve it.
 
Define the ##\mu :SO \left(3\right)/SO\left(2\right) \rightarrow S^2## by

$$\mu \left[ A \right] = A \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} .$$

You have to show that:

1) ##\mu## is well-defined, i.e., independent of the A used to define the coset [A]

2) ##\mu## is one-to-one

3) ##\mu## is onto.
 
Thread 'Help with Time-Independent Perturbation Theory "Good" States Proof'
(Disclaimer: this is not a HW question. I am self-studying, and this felt like the type of question I've seen in this forum. If there is somewhere better for me to share this doubt, please let me know and I'll transfer it right away.) I am currently reviewing Chapter 7 of Introduction to QM by Griffiths. I have been stuck for an hour or so trying to understand the last paragraph of this proof (pls check the attached file). It claims that we can express Ψ_{γ}(0) as a linear combination of...

Similar threads

Replies
29
Views
3K
Replies
1
Views
2K
Replies
2
Views
2K
Replies
1
Views
3K
Replies
1
Views
4K
Back
Top