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

[quantum_smile]

Homework Statement

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

Homework Equations

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

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!

 

[George Jones]

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 $S^2$?

 

[quantum_smile]

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.

 

[George Jones]

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

$$S^2 = \left\{ \mathbf{v} \in \mathbb{R}^3 | \mathbf{v} \cdot \mathbf{v} = 1 \right\}$$

 

[dextercioby]

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.

 

[George Jones]

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.