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 $$γ(x)=x+γ$$ for $x \in\mathbb R^n$, $γ ∈ Γ$; the action is properly discontinuous, and determines the Riemann covering $p:ℝ^n→ℝ^n/Γ$.
Precision : The lattice $\Gamma$ is defined to be $$\Gamma = \left\{\sum_{j=1}^n \alpha^j v_j : \alpha^j \in \mathbb{Z}, j=1,\dots, n\right\}.$$
How is defined the inner product $g_p$ on $T_p \mathbb{R}^n/\Gamma$ at the point $p$?
Thanks!
The metric on $R^n/\Gamma$ is induced by the scalar product of $R^n$. More generally, let $(X,g)$ be a manifold $X$ endowed with a differentiable metric $g$, $G$ a subgroup of isometries which acts properly and freely on $X$, $X/G$ is a manifold and $g$ induces a metric on $X/G$ as follows:
Let $p:X\rightarrow X/G$ the projection (it is a covering), $x\in X/G, u,v\in T_xX/G$ let $y\in X$ such that $p(y)=x$, since $dp_y:T_yX\rightarrow T_xX/G$ is an isomophism, there exists $u',v'\in T_yX$ such that $dp_y(u')=u, dp_y(v')=v$. Write $g'_x(u,v)=g_y(u',v')$. The definition does not depend on the choices. If $p(z)=x,$ there exists $h\in G$ such that $h(y)=z$, $dp_z(dh_y(u'))=u, dp_z(dh_y(v'))=v$. We have $g_z(dh_y(u')),dh_y(v'))=g_{h(y)}(dh_y(u')),dh_y(v'))=g_y(u',v')$ since $g$ preserves the metric.
In the example here, $G$ is a group of translations thus preserves the Euclidean metric.