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

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In summary: By definition, it is the matrix that diagonalizes the determinant of the matrix [A] defined above.2) ##\mu## is one-to-one: It is the matrix that is equal to the product of the two matrices in the coset [A].3) ##\mu## is onto: By Definition of a Lie group, every matrix in the group is onto.
  • #1
quantum_smile
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Homework Statement


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

Homework Equations


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

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!
 
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  • #2
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##?
 
  • #3
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.
 
  • #4
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

$$S^2 = \left\{ \mathbf{v} \in \mathbb{R}^3 | \mathbf{v} \cdot \mathbf{v} = 1 \right\}$$
 
  • #5
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.
 
  • #6
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.
 

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

1. What is SO(3)?

SO(3) is a special orthogonal group in three-dimensional space. It consists of all rotations around the origin in three-dimensional Cartesian coordinates.

2. What is SO(2)?

SO(2) is a special orthogonal group in two-dimensional space. It consists of all rotations around the origin in two-dimensional Cartesian coordinates.

3. What is S^2?

S^2, also known as the 2-dimensional sphere, is a mathematical representation of the surface of a sphere. It is a two-dimensional manifold that can be defined as the set of points in three-dimensional space that are equidistant from a fixed point, known as the center of the sphere.

4. How is SO(3)/SO(2)=S^2 proved?

This proof relies on the fact that every element in SO(3) can be written as a product of two elements in SO(2). Therefore, the quotient group SO(3)/SO(2) is equivalent to the set of all possible rotations around a fixed axis, which is equivalent to the surface of a sphere (S^2).

5. What is the significance of proving SO(3)/SO(2)=S^2?

This proof is significant in that it shows the connection between two seemingly different mathematical objects, namely the special orthogonal groups in two and three dimensions, and the surface of a sphere. It also has practical applications in fields such as physics, where rotations in three-dimensional space can be represented by elements in SO(3), and rotations in two-dimensional space can be represented by elements in SO(2).

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