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Taking the advice of Mariano Suárez-Alvarez, I moved this question from MO to MSE:


Let me introduce twisted (discrete) tori:

Consider the Cartesian graph product $\mathcal{C}_n = C_n \times P_m$ with $C_n$ the cycle with $n$ nodes, and $P_m$ the path with $m$ nodes. $\mathcal{C}_n$ is a discrete version of a cylinder, with two opposite "circles" (copies of $C_n$) as its top and bottom faces. By construction the nodes of these two circles are in a natural 1:1 correspondence (connected by unique shortest paths).

Now form the twisted tori over $\mathcal{C}_n$ by gluing these two circles. Gluing means: mapping the nodes of the two circles to each other respecting their cycle structure. The set of these "gluing maps" makes up the dihedral group $D_n$. There are $\lfloor n/2 \rfloor + 1 + 1$ conjugacy classes of $D_n$ as can be seen from the cycle graph of $D_n$. Thus there are $\lfloor n/2 \rfloor + 1 + 1$ isomorphism classes of twisted tori over $\mathcal{C}_n$.

One of them is a discrete version of the usual torus, and corresponds to $\text{id} \in D_n$. Another one is a discrete version of the Klein bottle, and corresponds to the orientation-reversing maps in $D_n$ (which are all conjugate). The other $\lfloor n/2 \rfloor$ ones correspond to proper "twists"

[Side note: In the special case $n=2$, it is not the discrete Klein bottle, but the discrete Möbius strip.]

The tori with order-preserving gluing maps are embeddable in an orientable surface, namely the torus, i.e. they are toroidal graphs. The torus with order-reversing gluing map(s) is not toroidal. All tori have as a characteristic the order of their gluing map $g \in D_n$. This order - roughly said - counts the number of times one has to "go around" on the torus before one gets back to the starting point.

[Side note: It seems worth noting that there is nothing like a "visible seam" where the gluing took place.]

Now to my question I already tried to ask here and here and here:


Can this approach been transferred to the continuous case, and what does remain?

In this case one starts with the cylinder $S_1 \times [0,1]$, and glues its ends by the maps of the continuous analogue of the dihedral group, i.e. the orthogonal group $O(2)$ which represent all the possible (order-preserving and reversing) "twists".

What happens to the many different isomorphism classes of discrete tori in the continuous case? Do they all collapse into one (resp. two) homeomorphism classes, and that's it?

The other way around: Which distinguishing geometrical (metric, topological) property of a twisted torus corresponds to the order of its gluing map $g \in O(2)$, at least when it is finite?

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