An $n$-dimensional Euclidean lattice $L$ is called of $\textbf{Voronoi's first kind}$ if it satisfies the $\textbf{obtuse superbasis}$ condition: There exist $x_0, \dots, x_n \in L$ such that
$1) x_1, \dots, x_n$ is a basis of $L$
$2) x_i \cdot x_j \leq 0$, unless $i=j$
$3) \sum_{i=0}^n x_i = 0$
Now I am reading this paper: The spine which was no spine.
The main result is the following: It uses the identification of the space $S_n = SO_n(\mathbb{R})$\ $SL_n(\mathbb{R})$ of (equivalence classes of) full-rank $n$-dimensional volume $1$ lattices up to rotation with the Teichmüller space $T_n$ of flat metrics of unit volume on the torus $\mathbb{T}^n = \mathbb{R}^n/\mathbb{Z}^n$.
Moreover, it identifies:
a)The $\textbf{systole} \operatorname{syst}(\rho)$ of a point $\rho \in T_n$, which is the length of the shortest homotopically essential geodesic in the flat torus $(\mathbb{T}^n, \rho)$, with the function $f:S_n \to (0, \infty), f(A) = \min_{v \in \mathbb{Z}^n, v\neq 0}|Av|$ which gives the length of the shortest vector in $A$.
b)The set $S(\rho)$ of homotopy classes of geodesics in $(\mathbb{T}^n, \rho)$ with length $\operatorname{syst}(\rho)$ with the set of shortest vectors of an element $A \in S_n$.
c)The set $X \subseteq T_n$ consisting of those points $\rho$ with the property that $S(\rho)$ generates a finite index subgroup of $\pi_1(\mathbb{T}^n)$ with the set of well-rounded lattices in $S_n$, that is, those lattices whose set of shortest vectors is a basis for $\mathbb{R}^n$.
to prove that $X$ is a minimal contractible deformation retraction of $S_n$ (that is, there does not exist a strict subset $Y \subset X$ with the above properties).
Now, I am a bit new in lattice theory in general, and especially in identifying lattice concepts with their respective diff. geometric counterparts, so is there a way to translate lattices of Voronoi's first kind into the diff. geometric context described above?
