Applications of flat submanifolds to other fields of mathematics
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
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.