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Developable surfaces in $\mathbb{R}^{3}$ have lots of applications outside geometry (e.g., cartography, architecture, manufacturing).

I am a curious about potential or actual applications to other fields of mathematics and science of flat submanifolds of $\mathbb{R}^{d}$, where $d >3$.

By flat I mean locally isometric to Euclidean space.

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    $\begingroup$ "of course" is not necessary. For instance, some people refer to "flats" as submanifolds that are both flat and totally geodesic, which is also of interest, while you don't assume totally geodesic. $\endgroup$ – YCor Feb 19 at 15:02
  • $\begingroup$ @YCor Thanks for the feedback $\endgroup$ – MK7 Feb 19 at 17:25
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  1. Crystallographic groups define flat compact manifolds and they are used to describe symmetries of crystals.

  2. Flat tori are used in computational physics and chemistry: if you want to investigate dynamics, say of a gas and and for computational reasons you can only consider 1000 particles, you cannot place the particles in $\mathbb{R}^3$ because they would escape. The trick is to place the particles in $\mathbb{S}^1\times \mathbb{S}^1\times \mathbb{S}^1$ which is represented as a "periodic" cube: if a particle leaves a cube through one side, in enters the cube on the opposite side.

  3. Math and art: By the famous Nash-Kuiper theorem, a flat torus $\mathbb{S}^1\times \mathbb{S}^1$ does admit a $C^1$ isometric embedding into $\mathbb{R}^3$. This is a very surprising result. There have been sculptures showing this embedding and you can see it on youtube: https://www.youtube.com/watch?v=RYH_KXhF1SY

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  • $\begingroup$ (2) I do not see how the flatness of the embedding of $S^1 \times S^1 \times S^1$ in $\mathbb{R}^6$ is used in the computation you are mentioning. It seems to me that you are only using the description of the $3$-torus as a topological identification space of the $3$-cube. $\endgroup$ – Francesco Polizzi Feb 20 at 7:30
  • $\begingroup$ @FrancescoPolizzi What you use is $\mathbb{R}^3/\mathbb{Z}^3$ and that is a flat torus isometric with the product of $\mathbb{S}^1$. $\endgroup$ – Piotr Hajlasz Feb 20 at 7:54
  • $\begingroup$ Thanks for the answer. Could you maybe indicate me some references for point (1) and (2)? $\endgroup$ – MK7 Feb 20 at 8:52
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    $\begingroup$ For crystallographic groups, just google. There is plenty of literature available and I don't know what would be the best source from your perspective. Regarding (2) I do not know the literature. My daughter does computational chemistry and this is what I learned from her. $\endgroup$ – Piotr Hajlasz Feb 20 at 9:11
  • $\begingroup$ OK thanks. From a brief search it seems to me that crystallographic groups are indeed flat manifolds, not submanifolds (of course they can be embedded isometrically, but I do not see any reason for doing so) $\endgroup$ – MK7 Feb 20 at 9:26
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The torus $T$ can be embedded as a flat submanifold of $\mathbb{R}^4$, the so-called Clifford torus. It is possible to put infinitely many different complex structures on $T$, and by Poincaré-Koebe Uniformization Theorem the resulting complex curves (known as elliptic curves) have the structure of a $1$-dimensional group variety over $\mathbb{C}$, their group law being induced by the translations of their universal cover $\mathbb{R}^2$.

Reduction over $\mathbb{F}_p$ of elliptic curves defined over $\mathbb{Q}$ are extensively used in modern cryptography.

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    $\begingroup$ Flatness of the embedding is in no way relevant to the complex structure(s), and even the structure of a complex elliptic curve is somewhat distant from the structure of elliptic curves over finite fields. Suggesting this somehow gives an application of flatness to cryptography is a big stretch. $\endgroup$ – Wojowu Feb 19 at 15:39
  • $\begingroup$ I do not agree, for at least two reasons. (1) The existence of a group law on $T$ comes from the corresponding additive group law in its universal cover $\mathbb{R}^2$, and the universal cover is such because of the flat metric (Poincare'-Koebe). (2) Many elliptic curves used in criptography are obtained by reducing modulo $p$ curves defined over the integer numbers, and after all the whole theory of elliptic curves over $\mathbb{F}_p$ (and so the corresponding criptosystems) came by analogy with the archetipal analytic construction. $\endgroup$ – Francesco Polizzi Feb 19 at 15:45
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    $\begingroup$ (1) There are various other reasons for deriving the group law, even using the universal covers - after all, non-flat tori have the same universal cover. (2) I don't disagree with either of the point, but for the most part the complex structure of rational curves is not used when we reduce them modulo a prime, and the analogy you mention doesn't give us any direct consequences of flatness to curves over finite fields. $\endgroup$ – Wojowu Feb 19 at 15:58
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    $\begingroup$ But Uniformization Theorem tells us that a closed surface is uniformized by the plane if and only if it admits at least one metric of constant curvature $0$. So, the existence of a flat embedding is crucial to have $\mathbb{R}^2$ as universal cover, and so the group law. In fact, hyperbolic curves are not group varieties, and not even topological groups (for instance because their fundamental groups is not abelian). $\endgroup$ – Francesco Polizzi Feb 19 at 16:03

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