Hopf Tori in S^3

Isometrically embedding flat tori into Euclidean space via the Hopf fibration.

In mathematics one grows accustomed to the fact that identifying opposing sides of a square yields a torus.
But the identification with the familiar rotation torus formed in 3 dimensions by revolving a circle around an axis (and the shape of a standard bagel) is only true in the rubber sheet world of topology, not the more rigid world of geometry. For two shapes to be geometrically equivalent (isometric)

Indeed there is a simple argument that there are no isometric embeddings of a flat torus into 3 dimensional Euclidean space. The argument goes like this. Assume you have some embedded torus; then its compact so it fits inside of some large sphere centered at the origin. Shrink this sphere until it first comes in contact with the torus. At the point of contact the sphere and torus are tangent, and the fact that the torus lies completly within the sphere implies the curvature of the torus is greater than or equal to the curvature of the sphere at that point. But of course, the sphere has positive curvature - so this point on the torus does as well: its not flat!

(Experts will note requiring the surface to be twice differentiable - so that curvature is defined - is crucial, as Nash provides fractal counterexamples! And also that this argument makes no use of the fact our surface is a torus: it applies to all sufficiently smooth compact surfaces in $\mathbb{R}^3$).

What about higher dimensions? Can we have flat tori there? The Nash embedding theorem guarantees that every compact Riemannian $n$ manifold can be isometrically embedded in $n(3n+11)/2$ dimensional Euclidean space, so flat tori must embed in $\mathbb{R}^{17}$. (Its good to pause and think about what goes wrong with the previous argument about shrinkwrapping with spheres, in higher dimensions). But this is just an upper bound: its certainly possible a shape also embeds in a smaller dimension (the round sphere for instance, embeds isometrically in $\mathbb{R}^3$ as well as $\mathbb{R}^{17}$).

For the particular case of the square torus, its also relatively easy to confirm one can do much better: through direct calculation one can confirm the following is an isometric embedding from $[0,2\pi]\times[0,2\pi]$ onto a toridal surface in 4 dimensional space

$(s,t)\mapsto \begin{pmatrix}\cos s\\ \sin s \\ \cos t \\ \sin t\end{pmatrix}$

(Check this is an isometry by computing the pullback metric from $\mathbb{R}^4$ to the $(s,t)$ plane, and showing it agrees with the usual metric $ds^2+dt^2$).

This answers the question for the square torus, but now that we’ve been concerned with geometry we must confront the fact that there are many, many geometrically distinct flat tori. The full argument is outside the scope of this short post, but here’s the cliffnotes. Any parallelogram can be rolled up into a flat torus, and any two ways of cutting a flat torus into a parallelogram differ by a change of basis: the edges of one are integer linear combinations of the edges of the other. Because any two such parallelograms are the same area this change of basis is an element of $GL(2,\mathbb{Z})$. And as the linearly independent sides of a paralleogram uniquely determine an element of $GL(2,\mathbb{R})$, we can identify the space of all possible flat tori with the quotient $GL(2,\mathbb{R})/GL(2,\mathbb{Z})$.