Define $h:\mathbb{R}\mathbb{P}^1 \rightarrow \mathbb{R}\mathbb{P}^1$ as $h([x_o:x_1])=[x_o ^2 : x_1 ^2 - x_0 x_1]$.
- Define charts around each point of $\mathbb{R}\mathbb{P}^1$.
- Write $h$ locally using these local coordinates.
- You should have written at least two local expressions for $h$, for different coordinate systems. What are the relations of these expressions in terms of transition functions?
I "know" all the definitions appearing in the question but do not know how to deal with them. Also I'm not that much familiar with projective spaces. Therefore any hints, "more than hint"s or solutions are surely appreciated.
Hint: What you need to know is that $\mathbb{RP}^1$ can be covered by two (affine) open subsets $U_i:=\{[x_0:x_1]| \; x_i \neq 0\}$ for $i=0,1$ which is isomorphic to $\mathbb{R}$ given by $U_0 \cong \mathbb{R}$ where $[x_0:x_1] \mapsto x_1/x_0$ and $U_1 \cong \mathbb{R}$ where $[x_0:x_1] \mapsto x_0/x_1.$ Note that the transition map from $U_0$ to $U_1$ is given by viewing both as two copies of $\mathbb{R}$ where they are attached on the overlap by $x\mapsto 1/x.$
Writing $h$ locally means to find an expression for $h|_{U_i}$ for each $i=0,1.$ Can you conclude from here now?