How can I understand the isomorphism $SO(3)\cong \mathbb{RP}^3 \cong S^3/\mathbb{Z}_2$ and compute the corresponding volumes?

Question

I want to understand the above isomorphisms $SO(3)\cong \mathbb{RP}^3 \cong S^3/\mathbb{Z}_2$. I seem to get some partial understanding but I miss a complete picture. For example I think that the last one is due to the fact that $\mathbb{RP}^3 \cong S^3/\mathbb{Z}_2$ is due to the fact that the projective space here is the sphere with antipodal points identified. Now, I also seem to get that $SO(3)$ is the 3-sphere after you fix the stabilizer, but I am not sure why the stabilizer here is $\mathbb{Z}_2$.

Now, I also want to compute the corresponding volumes of these manifolds and make sure they coincide (as they should). How can I do this? I am especially confused on how to calculate the volume of $\mathbb{RP}^3$.

Possible duplicate of Proving algebraically that $\mathbb RP ^3\cong SO(3,\mathbb R)$ — Dietrich Burde, Feb 16 '16 at 12:17
For a proof also see here, Marsden and Ratiu. For the second part see here. — Dietrich Burde, Feb 16 '16 at 12:22
The isomorphism $\mathbb{S}^3/\mathbb{Z}_2≅SO(3)$ you can find it here (http://files.umwblogs.org/blogs.dir/4710/files/2010/10/honors_triola.pdf). Also, the second isomorphism is just the action via the antipodal map of the group $\mathbb{Z}2$ on $\mathbb{S}^3$ whose quotient space is the projective space $\mathbb{RP}^3$. — Christos, Feb 16 '16 at 15:18
Note that in order to calculate volumes independently, you need to specify metrics (e.g. scalings for homogeneous metrics) independently; compare Diameter of the Grassmannian. — Andrew D. Hwang, Feb 19 '16 at 22:51

score 0 · Answer 1 · answered Feb 16 '16 at 14:34

Focusing on why $\mathbb{Z}_2$ since it seems you are comfortable with the rest, here's how I would view it.

In geometric algebra we can make a rotor by multiplying two unit vectors. We have $$R = \mathbf{uv} = e^{\theta\mathbf{B}} = \cos\theta + \mathbf{B}\sin\theta\quad\text{ where }\mathbf{B} = \frac{\mathbf{u}\wedge\mathbf{v}}{|\mathbf{u}\wedge\mathbf{v}|^2}$$ i.e. $\mathbf{u}\wedge\mathbf{v}$ normalized. Now to actually apply the rotation we use the versor product $$T(\mathbf{v}) = R\mathbf{v}\widetilde{R}$$ where $\widetilde{R} = e^{-\theta\mathbf{B}} = \cos\theta - \mathbf{B}\sin\theta$. However, the action of this transform is to rotate $\mathbf{v}$ through $2\theta$ radians in the $\mathbf{u}\wedge\mathbf{v}$ plane. We can just immediately note that $-R$ does the same thing as $R$. If we fix a reference vector $\mathbf{v}$ in the definition of the rotor, then we have a one-to-one correspondence between unit vectors and rotors. Thus, $S^3/\{-1,1\}$.

To calculate the volume you're ultimately going to integrate and to integrate you're ultimately going to parameterize. The manifolds are non-orientable, but in this case that just means sticking to only one hemisphere, say the northern. The equator would be awkward if it wasn't a set of measure zero and thus irrelevant to the value of an integral. You can formulate it as an integral over all points above a certain latitude, then just take the limit as the latitude approaches the equator. The differences between calculating the volumes of the spaces are only in how to map the points of the spaces to parameters. Once we've parameterized one, given diffeomorphisms between the others provides a way to parameterize them as well. Of course that somewhat undermines independent verification.

How can I understand the isomorphism $SO(3)\cong \mathbb{RP}^3 \cong S^3/\mathbb{Z}_2$ and compute the corresponding volumes?

1 Answers1