I'm computing measures on quotient spaces of SO(3) and I have questions about the integration of these measures.
I'm parameterizing SO(3) by the Euler angles
$R(\phi,\theta,\psi) = R_z(\phi)R_y(\theta)R_z(\psi)$
with the ordinary rotation matrices $R_y$ and $R_z$ and $0 \leq \phi \leq 2\pi, 0 \leq \theta \leq \pi, 0 \leq \psi \leq 2 \pi$.
By computing the matrix $R^{-1} dR$, we get all 3 Maurer-Cartan forms:
$\omega_{12} = -\cos \theta\, d \phi - d\psi\\ \omega_{13} = \sin \theta \sin \psi\, d \phi + \cos \psi\, d \theta\\ \omega_{23} = \sin \theta \cos \psi\, d \phi - \sin \psi\, d \theta$
(indices according to position in the matrix $R^{-1} dR$).
The Haar measure of SO(3) can be constructed from the Maurer-Cartan forms
$d \mu = \omega_{12}\wedge \omega_{13} \wedge \omega_{23} = \sin \theta\, d \psi \wedge d \phi \wedge d \theta$
which I can integrate to get the "volume" of SO(3):
$\int d \mu = \iiint \sin \theta \, d \psi d \phi d \theta = 8\pi^2$
Now I take as a 1-dim subgroup H, all elements of SO(3) that leave the $z$-axis invariant. Thus, H is characterized by $\theta = 0, \phi = 0$. This leads to $d \theta = 0, d \phi = 0$ which is equivalent to $\omega_{13} = 0, \omega_{23} = 0$.
The measure on SO(3)/H can then be constructed by
$d \mu_z = \omega_{13} \wedge \omega_{23} = \sin \theta \, d \theta \wedge d \phi$
and I can integrate this to get the "volume" of SO(3)/H:
$\int d \mu_z = \iint \sin \theta \, d \theta d \phi = 4\pi$
(this seems to be correct, because I think that $SO(3)/H \simeq S^2$)
When choosing other axis for the subgroup H (fixing two of the three angles), I come up with the following measures
$d \mu_x = \omega_{12} \wedge \omega_{13} = -\cos \theta \cos \psi \, d\phi \wedge d \theta - \sin \theta \sin \psi \, d\psi \wedge d \phi - \cos \psi \, d\psi \wedge d \theta\\ d \mu_y = \omega_{12} \wedge \omega_{23} = \cos \theta \sin \psi \, d\phi \wedge d \theta - \sin \theta \cos \psi \, d\psi \wedge d \phi + \sin \psi \, d\psi \wedge d \theta$
but I don't know how to integrate them.
I guess they are constructed correctly, because since $d (d\mu_x) = 0, d(d\mu_y) = 0$ they are integrable. I also guess that the result is the same as for $d\mu_z$.
A similar problem arises when I try to do this computation with angle-axis-coordinates for SO(3).
I appreciate any help or suggestions.