# How to show that $SO(3)$ and $\mathbb{R}P^3$ are dffeomorphic?

Pardon me for repeating a question, but I am not able to justify one aspect of this proof that $$SO(3)$$ and $$\mathbb{R}P^3$$ are diffeomorphic, and I was hoping that someone could clarify it for me.

Claim: $$SO(3)$$ is diffeomorphic to $$\mathbb{R}P^3$$.

Proof. Let $$D^3$$ be the closed unit ball centered at the origin in $$\mathbb{R}^3$$. Map each nonzero $$x \in D^3$$ to the matrix in $$SO(3)$$ corresponding to clockwise rotation by $$\pi|x|$$ of $$\mathbb{R}^3$$ around the directed axis determined by $$x$$. Also map the origin to the identity matrix.

This is a surjective map because every nonidentity element of $$SO(3)$$ corresponds to a clockwise rotation of $$\mathbb{R}^3$$ about some directed axis passing through the origin. It is injective on the interior of $$D^3$$, and two-to-one on the boundary of $$D^3$$ with antipodal points mapping to the same element of $$SO(3)$$.

This map is also continuous, so we obtain a continuous bijection from $$D^3 / \sim$$ to $$SO(3)$$, where $$\sim$$ is the equivalence relation given by identifying antipodal points on the boundary of $$D^3$$. Since $$\mathbb{R}P^3 = D^3 / \sim$$ is compact and $$SO(3)$$ is Hausdorff, this mapping is in fact a homeomorphism.

This map can also be shown to be a diffeomorphism. $$\tag*{\blacksquare}$$

The problem is I am unable to properly justify that it is a diffeomorphism. If $$f : D^3 \to SO(3)$$ is the map under discussion, then we have $$f(x) = A(x/|x|, \pi|x|) \quad \text{ for all nonzero } x \in D^3,$$ where $$A(v,\theta)$$ is the matrix in $$SO(3)$$ corresponding to clockwise rotation of $$\mathbb{R}^3$$ by $$\theta$$ about the directed axis determined by the unit vector $$v \in \mathbb{R}^3$$. If $$R_x(\theta)$$, $$R_y(\theta)$$ and $$R_z(\theta)$$ are the clockwise rotations by $$\theta$$ about the $$x$$-, $$y$$- and $$z$$-axes, respectively, then we have $$A(v,\theta) = R_z(\alpha) R_y(\beta)^T R_z(\theta) R_y(\beta) R_z(\alpha)^T,$$ where $$v = (\cos \alpha \sin \beta, \sin \alpha \sin \beta, \cos \beta)$$ in spherical coordinates. This is taken from @Randall's answer to this question.

With this description, it is easy to see that $$f$$ is continuous, because the entries of $$A(v,\theta)$$ are continuous (and, in fact, smooth) functions of $$\theta$$, $$\alpha$$ and $$\beta$$ (and hence of $$v$$ too). Thus, $$f$$ is a smooth map when we view $$SO(3) \subset M_3(\mathbb{R}) \underset{\text{diff.}}{\cong} \mathbb{R}^{3^2}$$, and in particular it is continuous.

Let $$g : \mathbb{R}P^3 \to SO(3)$$ be the map induced by $$f$$ on the quotient space. To show that it is a diffeomorphism, it only remains to show that $$g^{-1}$$ is smooth.

The only way I know to show that a map between smooth manifolds is smooth is to use the definition: take smooth charts on both sides and compose to get a map between subsets of Euclidean spaces, and then check that this is smooth. Here, I have a problem. I know that $$SO(3)$$ is a smooth manifold because I have shown that the $$O(3)$$ is the inverse image of a regular value, and $$SO(3)$$ is a connected component of $$O(3)$$. I don't have an explicit description of a smooth atlas on $$SO(3)$$ and so I am unable to show that $$g^{-1}$$ is smooth.

Can someone help me complete this last step of the problem?

• Try extending this map to all of $\Bbb R^3$ by the sane formula and showing that on the ball of radius less than $2\pi$, it is a submersion. – user98602 Nov 11 '18 at 3:50
• @MikeMiller how would that help, can you elaborate? – Brahadeesh Nov 11 '18 at 4:38