In the book Eigenvalues in Riemannian Geometry of Isaac Chavel page $28$, I have some questions related to the resolution of the spectrum of the tori.
The lattice acts on $\mathbb{R}^n$ by $$\gamma(x)=x+\gamma$$ for $x \in \mathbb{R}^n$, $\gamma \in \Gamma$; the action is properly discontinuous, and determines the Riemann covering $p : \mathbb{R}^n \to \mathbb{R}^n/\Gamma.$
- When it says "the action determines the Riemannian Covering $p : \mathbb{R}^n \to \mathbb{R}^n/\Gamma$", does it means $p(x)= x + \Gamma = \{\gamma(x) : \gamma \in \Gamma\}$?
- Does $\mathbb{R}^n/\Gamma$ is a Riemann manifold? If so, why?
- $p$ is a Riemannian covering if it is differentiable covering and local isometry. Could anyone be able to prove that part?