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.
 A: *

*Crystallographic groups define flat compact manifolds and they are used to describe symmetries of crystals. 

*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.

*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 
A: 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.  
