We know that there is a Riemannian covering map $p:S^n \rightarrow \mathbb{R}P^n$ and I'm working with some foliation problem, which ended up in trying to find a Riemannian covering map of $\mathbb{C}P^n$, but I don't know any Riemannian cover of this space.
I would like to find some references to this subject, thanks in advance.
Since $\mathbb{C}P^n$ is simply connected, it has no nontrivial covers (that is, any covering space if $\mathbb{C}P^n$ is just a disjoint union of copies of $\mathbb{C}P^n$). In general, if $X$ is a connected space that is reasonably nice (e.g., a manifold), then connected covering spaces of $X$ are classified by subgroups of the fundamental group $\pi_1(X)$. You can find details in any introductory text on algebraic topology.
(Note that if $X$ has some additional local structure, such as that of a Riemannian manifold, then for any covering space $p:Y\to X$ you can uniquely lift that local structure to $Y$ such that $p$ preserves it, since $p$ is locally a homeomorphism. So, for instance, classifying Riemannian covering spaces of $\mathbb{C}P^n$ is the same as just classifying ordinary topological covering spaces of $\mathbb{C}P^n$.)