How Is the Volume of SU(2) Calculated?

[eko_n2]

Homework Statement

Compute the volume of the group SU(2)

Homework Equations

Possibly related: in a previous part of the problem I showed that any element

g = cos(\theta) + i \hat{n} \cdot \vec{\sigma}sin(\theta)

The Attempt at a Solution

How do I compute the infinitesimal volume element dV?

 

[fzero]

There are a few ways to do this, but it depends on where you are coming from, type of course, etc. If you know that $SU(2)=S^3$, you can use your parameterization to define coordinates on the 3-sphere as imbedded in $\mathbb{R}^4$. If you know some more geometry, you could construct the Lie-algebra valued 1-form $\omega = g^{-1} dg$ and then use that to construct a volume form.

In order to be more helpful (within the rules of the forum that say only give hints), you'd have to be more precise about what you already know.

 

[andrien]

I have heard somewhere(I don't remember where) that
U(n)/U(n-1)=S^2n-1,where Sⁿ is n dimensional sphere embedded in Rⁿ⁺¹.With U(1)=2∏ one can go on for simple manipulation to get U(n).But I can not go for any book which contain any information about this.May be someone can provide any reference.