Smooth variety of lattices

In the previous work of the authors, the foundations of the theory of smooth manifolds of number-theoretic lattices were laid. The simplest case of one-dimensional lattices was considered.
This article considers the general case of multidimensional lattices.
Note that the geometry of the metric spaces of multidimensional lattices is much more complex than the geometry of ordinary Euclidean space. This is evident from the paradox of the non-additivity of the length of a segment in the space of shifted one-dimensional lattices. From the presence of this paradox it follows that there is an open problem of describing geodesic lines in spaces of multidimensional lattices, as well as in finding a formula for the length of arcs of lines in these spaces. Naturally, it would be interesting not only to describe these objects, but also to obtain a number-theoretic interpretation of these concepts.
A further direction of research could be the study of the analytical continuation of the hyperbolic zeta function on spaces of multidimensional lattices. As is known, the analytical continuation of the hyperbolic zeta function of lattices was constructed for an arbitrary Cartesian lattice. Even the question of the continuity of these analytic continuations in the left half-plane in lattice space has not been studied. All of these, in our opinion, are relevant areas for further research.