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.



Applications of flat submanifolds to other fields of mathematics
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>


