Basic questions about the flat torus

Question

I am trying to undesrtand a bit better the geometry of the flat torus $T^n=\mathbb{R}^n/{\mathbb{Z}^n}$ and during this process some question that I would like to solve have arising:

Regarding the metric I have seen this post Metric of the flat torus . However, it is not clear to me what the answer really means. Does it means that the only riemanian metric in the torus invariant under traslantions is indeed $d(x+\mathbb{Z}^n,y+\mathbb{Z}^n)=\min\left\lbrace d_E(x+z_1,y+z_2): z_1,z_2\in \mathbb{Z}^n\right\rbrace$?, where $d_E$ denotes the euclidean distance in $\mathbb{R}^n$.

EDIT Thanks to the comment of @Artic Char I have seen that this is the distance that is usually considered in the flat torus. However what I don't quite see is what should be the matrix of this metric? Does anyone know if it's easy to write, at least in $T^2$?

Given $f:T^n\to\mathbb{R}$, we consider $\int_{T^n}|\nabla f|^2$. I was wondering if, in order to make sense of the gradient, can we use the identification of functions defined on the torus with the $2\pi$-periodic functions in $\mathbb{R}^n$ to consider the usual gradient or we have to define it in a different way. On the other hand, when we write $|\cdot|$, is this norm depending of the metric above?

Just to be sure, is your $g$ supposed to be a Riemannian metric? — Arctic Char, Dec 06 '23 at 02:03
Then it should be different from the metric in the definition of metric space? Your notation $g(x+\mathbb{Z}^n,y+\mathbb{Z}^n)=\min\left\lbrace d_E(x+z_1,y+z_2): z_1,z_2\in \mathbb{R}^n\right\rbrace$ looks quite confusing, if $g$ is indeed a Riemannian metric. — Arctic Char, Dec 06 '23 at 02:06
For 1, there are more translation invariant (Riemannian metric) for example, just choose $g_{ij}(x) = a_{ij}$ for some fixed positive definite matrix $A = (a_{ij})$, then the corresponding Riemannian distance function is translation invariant — Arctic Char, Dec 06 '23 at 02:18
For 2, one you have a Riemannian metric you can define $\nabla f$ as a vector field, and $|\nabla f| = \sqrt{g(\nabla f, \nabla f)}$ so $|\nabla f|$ depends on the Riemannian metric chosen. — Arctic Char, Dec 06 '23 at 02:19
Thanks for your answers. So when we consider the flat torus, which is the usual metric? — mejopa, Dec 06 '23 at 02:20
For the question you ask in your edit, yes the matrix of the metric (with respect to the canonical coordinate sytem) is easy to write. It is just the identity matrix. — Paul Sinclair, Dec 06 '23 at 22:07

Basic questions about the flat torus

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