Give an explicit atlas for the manifold $\mathbb{R}P^3$ which is defined as the quotient of the three-sphere by the antipodal mapping.

Question

I'd like to know if there is an explicit atlas for the manifold $\mathbb{R}P^3$ which is defined as the quotient of the three-sphere by the antipodal mapping.

Thanks.

Answer 1

Take first an explicit atlas for $\mathbb{S}^3 $ $\phi_{i, \pm}\colon U_{i, \pm} \to D_3$ where $U_{i, \pm} = \{ x \in \mathbb{S}^3 \ | \ \pm x_i >0\}$, with $\phi_{i,\pm}$ the projection onto the components with index not $i$. This is an atlas for $\mathbb{S}^3 $. It's strongly recommended to check for oneself.

Note now that the restriction of the canonical projection $\mathbb{S}^3\to \mathbb{R}P^3$ is injective. Take $\bar U_{i,\pm}$ the image of this set. It will be an open subset in $\mathbb{R}P^3$. Consider the map $\bar \phi_{i,\pm} \to D_3$ to be the composition of the $\phi_{i, \pm}$ with the inverse of the restriction of the projection $\mathbb{S}^3\to \mathbb{R}P^3$.

One checks that we get again an atlas. The only thing that we are using is that the restriction $\mathbb{S}^3\to \mathbb{R}P^3$ is injective. (the identifying is done in a structured way). The coordinate changes are OK since they are also coordinate changes for the atlas of $\mathbb{S}^3$.

The same argument works for any $\mathbb{S}^n$. It is useful to look at the picture for the cases of $\mathbb{S}^1$ and $\mathbb{S}^2$.

Answer 2

For any $n$, real projective $n$-space $\Bbb P^n(\Bbb R)$ is actually isomorphic to the quotient $$ \frac{\Bbb R^{n+1}\setminus\{0\}}\sim\qquad \text{where $u\sim v$ iff $u=\lambda v$ for some $\lambda\in\Bbb R^\times$.} $$ Then to every point $P\in\Bbb P^n(\Bbb R)$ one can associate its homogeneus coordinates $P=[u_0,u_1,...,u_n]$ which are defined up to a scalar (they are the coordinates of any $\tilde P\in\Bbb R^{n+1}$ representing $P$).

Then the standard atlas is made up by the $n+1$ charts $$ U_i=\{u_i\neq 0\}. $$ One sees that $U_i\simeq\Bbb R^n$ simply normalizing $u_i=1$ in the homogeneous coordinates of its points.

At this point, it is a straightforward exercise to write the transition functions.