This is a simple question, but I can't figure out the name for this class of topological space.
Say you start with the affine space $\mathbb{R}^n$ for finite n, and equip it with a metric.
Now, say you impose the equivalence relation generated by the relation such that for any two points $p, q$, $p \sim q \iff p-q \in V$, where V is a set of linearly independent vectors and $|V| \leq n$. This gives you a new topological space which can be thought of as a topological quotient space of the original (or also thought of as an orbit space). I will notate this space $\mathbb{R}^n/V$, and note that this new space also naturally inherits a metric from the old one via the quotient metric.
- Example: if you start with $\mathbb{R}^2$ and do this with $|V|$ = 1, you end up with the cylinder, and if you do it with $|V|$ = 2, you end up with the flat torus.
Consider the set of all spaces one can obtain in this way; by starting with some finite-dimensional $\mathbb{R}^n$ and modding by some set of vectors $V$. Is there a name for this particular class of metric space? It seems like this would be a subset of the set of "Finsler manifolds," but I don't know if there's anything more specific than that.
Also, if it makes this question easier to answer, you can ignore the metric and just focus on the particular set of topological spaces that are constructible in this way; does this set have a name?
EDIT: as mentioned, this is the same as the set of all spaces one can generate by $\mathbb{R}^a \times T^b$. Do these spaces have any sort of canonical name? They're one way to generalize a cylinder, but not the only way.
Your notion is equivalent to the following: Start with your subset $V$ and generate the additive subgroup $\Gamma< R^n$ by this subset. Now, form the quotient $R^n/\Gamma$ as in the theory of covering spaces: You identify points $p, q\in R^n$ whenever there exists $\gamma\in \Gamma$ sending $p$ to $q$. The quotient space is diffeomorphic to $M=R^{n-k} \times T^{k}$, where $k$ is the rank of the free abelian group $\Gamma$, $T^k$ is the $k$-dimensional torus. Note that $\Gamma$ is the group of automorphisms of the covering map $R^n\to M$. I am not sure there is a standard name for such spaces. See http://en.wikipedia.org/wiki/Flat_manifold for references.