Bianchi and Hopf (Annuli V)

When you take a torus in Euclidean space, it will always have points of positive curvature and points of negative curvature, but the average of the curvature will be 0.

This limitation of Euclidean space disappears in the 3-sphere. Luigi Bianchi more or less completely classified all flat tori in the 3-sphere in the late 19th century, paving the way to global differential geometry.


Simple examples can be constructed using the Hopf fibration: If you take the preimage of a simple closed curve in the 2-sphere under the Hopf fibration, you obtain a torus in the 3-sphere that is intrinsically flat.

When you start with a circle in the 2-sphere, the result will be a torus of revolution (or a Dupin cyclide), which corresponds to a rectangular torus. If you start with more general curves like the spherical cycloids that I have used here, you will get tori that appear twisted.
This is because these tori will correspond to non-rectangular tori, as can be verified using an elegant formula due to Ulrich Pinkall. The side view below gives access to the warped core of these tori.

The image below shows the view from somebody standing inside such a Hopf torus, with an almost perfect mirror as a surface, and four differently colored light sources. This comes pretty close to my ideal of abstract 3-dimensional art. If you know what you are looking at, you can discern the same warped core as up above, and its reflections on the warped outer parts of the torus.


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