Quotient Riemann metric If $M$ is a Riemann manifold and $G$ a group that act discontinuously in M. How can I induce a Riemann metric on the quotient $M/G$? I tried some things, but I didn't move.
And, how can I induce a metric in the projective space? Here my attempts were: $\mathbb{P}^{n}\left(\mathbb{R}\right) = \mathbb{S}^{n}/E$, where E is the antipodal equivalence relation. So $A: \mathbb{P}^{n}\left(\mathbb{R}\right) \rightarrow \mathbb{S}^{n}$ such that $A\left([p]\right) = p$, I would like to prove that is smooth and is a immersion, with this I could induce a metric in $\mathbb{P}^{n}\left(\mathbb{R}\right)$, but I don't know how do this.
Thanks in advance.

$\pi : S^n\rightarrow X:=\mathbb{P}^n$ is covering If we give a suitable metric on $X$ then $\pi$ is locally isometric : 
For $B_r([p]),\ [p]\in X$ and $r$ is small then $$ d_X([a],[b]):=d_{S^n}(a,b) $$ for $[a],\ [b]\in B_r([p])$ 
