Related to the book Eigenvalues in Riemannian Geometry by Isaac Chavel page $30$, we have the proof of Weyl's asymptotic law for Flat Torus that $\text{Vol } \mathbb{T} = \text{Vol } P(v)$, but I can't explain why it is true. Could anyone be able to explain this relation?
Let $\{v_1, \dots, v_n\}$ be the basis of the lattice $\Gamma$. To each $v=\sum_{j=1}^n \alpha_j v_j$ in $\Gamma$ associated the parallelopiped $P(v)$ defined by $P(v) = \{x = \sum_{j=1}^n \beta_j v_j : \alpha_j < \beta_j < \alpha_j +1 : j= 1, \dots, n\}.$
Clarification : $\mathbb{T}$ is the torus here.
Thanks for the explanation!
This is not an "asymptotic law" but an immediate consequence of the very idea of ${\mathbb T}:={\mathbb R}^n/\Gamma$. The full space ${\mathbb R}^n$ is the "universal cover" of ${\mathbb T}$, and any parallelepiped spanned by $v_1$, $\ldots$, $v_n$ serves as a "fundamental domain", i.e., as a set of representatives for the points of ${\mathbb T}$. The volume of ${\mathbb T}$ then is equal to the volume of such a parallelepiped (by definition) and is equal to $\bigl|{\rm det}(V)\bigr|$, where $V$ is the $n\times n$-matrix having the components of the $v_k$ in its columns.