I encountered in my research on dynamical systems a problem, which considers for some $L>0$ on the $C\_n=[0,L]^n$ the set $\mathcal{C}\_n=\{(x\_1,\ldots,x\_n)\mid\exists j,k:\,x\_{i\_j}=x\_{i\_k}\}$. I am looking for an interpretation of $\mathcal{C}\_n$ on the $n$-torus, when imposing periodic boundary conditions on $C\_n$. For $n=2$ this is a simple closed curve which winds once around the torus. For $n=3$, I am already struggeling what $\mathcal{C}\_3$ "looks like". In particular, I am wondering how curves of the form $\gamma:t\mapsto x+t\, e\_j$, with $(e\_j)\_i=\mathrm{1}\_{i=j}$, intersect with $\mathcal{C}\_n$.


In general, literature recommendations on the $n$ torus are highly appreciated!



Curves on $n$-torus analogous to curve implied by diagonal in square for torus
I suppose I should add a caveat before turning my comment into an answer. In your definition, you have the $\exists j,k$ qualifier, which I believe means that $\mathcal C\_n$ is the union of $n \choose 2$ tori, i.e. one for every choice of $j \neq k$. If you allow $j=k$ then I suppose you would have $\mathcal C\_n = C\_n$, but I doubt you meant that. Finally, if you take the statement $\exists j,k$ out of your set definition, you would have a single torus. With that caveat, I'll expand my comment.


The observation follows from your equation $x\_{i\_j} = x\_{i\_k}$, this is basically stating your set is the graph of a function.


A slightly different perspective on a related idea. Any $n \times n$ matrix $A$ with integer coefficients and $\det(A) = \pm 1$ induces an affine-linear automorphism of $\mathcal C\_n$ (thought of as a torus).


Provided you are okay with that, notice that the matrix


$$\pmatrix{1 & 1 \\ 0 & 1}$$


sends $\{0\} \times [0,L]$ to $C\_2$.


i.e. we have turned the equation $x\_1=x\_2$ into the functional expression $(x\_1, x\_2) = (x\_2, x\_2)$, turning $\mathcal C\_2$ into the image of the function $x\_2 \longmapsto (x\_2,x\_2)$, or the graph of $x\_1(x\_2) = x\_2$, i.e. $x\_1$ as a function of $x\_2$ is equal to $x\_2$.


You can do the same with $C\_3$, writing it as the union of the graphs of the functions


$$\pmatrix{x\_1\\x\_2\\0} \longmapsto \pmatrix{x\_1 \\x\_2\\x\_2}$$
$$\pmatrix{x\_1\\0\\x\_3} \longmapsto \pmatrix{x\_1 \\x\_1\\x\_3}$$
$$\pmatrix{0\\x\_2\\x\_3} \longmapsto \pmatrix{x\_3 \\x\_2\\x\_3}$$


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