We have $\mathbb{R}^3$, and then the quotient $$\mathbb{R}\mathbb{P}^2=\mathbb{R}^3/\sim$$ where $x\sim y$ if $\exists k\in\mathbb{R}^*$ such that $x=ky$.
I have the open sets $U_1,U_2,U_3$ where $U_i=\{[p_1,p_2,p_3]\mid p_i\neq 0\}$ and the charts: $$\phi_i:U_i\to\mathbb{R}^2$$ given by $$\phi_1([p_1,p_2,p_3])=\left(\frac{p_2}{p_1},\frac{p_3}{p_1}\right)$$ $$\phi_2([p_1,p_2,p_3])=\left(\frac{p_1}{p_2},\frac{p_3}{p_2}\right)$$ $$\phi_3([p_1,p_2,p_3])=\left(\frac{p_1}{p_3},\frac{p_2}{p_3}\right)$$
so in this way we get a differentiable structure on it.
I want to know how, with this differentiable structure we can build a Riemannian metric on it. Would it be different from the metric induced by the first fundamental form of $S^2$ and the fact that the antipodal mapping is an isometry?