Applications of flat submanifolds to other fields of mathematics

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.

• "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. – YCor Feb 19 at 15:02
• @YCor Thanks for the feedback – MK7 Feb 19 at 17:25

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

• (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. – Francesco Polizzi Feb 20 at 7:30
• @FrancescoPolizzi What you use is $\mathbb{R}^3/\mathbb{Z}^3$ and that is a flat torus isometric with the product of $\mathbb{S}^1$. – Piotr Hajlasz Feb 20 at 7:54
• Thanks for the answer. Could you maybe indicate me some references for point (1) and (2)? – MK7 Feb 20 at 8:52
• 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. – Piotr Hajlasz Feb 20 at 9:11
• 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) – MK7 Feb 20 at 9:26

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.

• 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. – Wojowu Feb 19 at 15:39
• 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. – Francesco Polizzi Feb 19 at 15:45
• (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. – Wojowu Feb 19 at 15:58
• 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). – Francesco Polizzi Feb 19 at 16:03