Differentiable map between $S^2$ and $\Bbb{RP}^2$.

Question

Let $\Phi$ be the map sending a point $(x_0,x_1,x_2)\in S^2$ to it's equivalence class $[x_0,x_1,x_0]\in \mathbb{RP}^2$. The claim is that this map is a differentiable map from $S^2$ to $\Bbb{RP}^2$ given a certain choice of charts.

The question is: what is the best choice of charts on $S^2$ and $\Bbb{RP}^2$ to make explicit computations easy?

Answer 1

The usual choice of charts for $\Bbb R\Bbb P^n$ is given by the $n+1$ open sets $U_0,\dots,U_n$ defined by $$ U_i = \left\{ \,[x_0 : \dots : x_n] \,\middle|\, x_i\neq 0\,\right\}. $$ These all map bijectively to $\Bbb R^n$. For example $U_0\to \Bbb R^n$ is given by $[1:x_1:\dots:x_n] \mapsto (x_1,\dots,x_n)$.

I guess the best matching charts on $S^n$ should be the $2(n+1)$ open sets given by $x_i>0$ or $x_i<0$ for each $i=0,1,\dots,n$.

Answer 2

Try to use homogenous coordinates on $\mathbb{R}\mathbb{P}^2$ - that should lead you to a natural choice of charts on $S^2$ as well.

Answer 3

Let be $x: U \to S^2$ given by $x(u,v) = (u,v,(1 - u^2 - v^2)^{1/2})$ where $U = \{(u,v) ; u^2 + v^2 < 1\}$ .$y:R^2 \to RP^2$ given by $y(r,s) = [r,s,1]$. See that $x$ and $y$ are charts of $S^2$ and $RP^2$ such that $\Phi\circ x(U) \subset y(R^2)$. Therefore $$y^{-1}\circ \Phi \circ x (u,v) = (u,v)$$ Other case are analogous and thus $\Phi$ is smooth.