The Clifford torus shows up a lot in differential geometry in connection with minimal surfaces, for example in the Lawson's conjecture, the Oh's Conjecture, etc. It can be described as the following surface in $\mathbb C^2$: $$T = \left\{ (z_1, z_2) \in \mathbb C^2: |z_1| = |z_2| = \frac{1}{\sqrt 2}\right\}.$$

I'd like to know in what context it was introduced, and why it is called the Clifford torus. From the Wikipedia page, Clifford is a geometer, but it does not mention if he used this torus in any way.


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The Clifford torus was introduced by Clifford in 1873, not as embedded into $\mathbb{R}^4$ or $\mathbb{C}^2$, but first projectively and then intrinsically, by identifying the opposite sides of a flat parallelogram, a commonplace method in modern topology textbooks. The embedding into $\mathbb{R}^4$ first appeared in Killing's Die Nichteuklidischen Raumformen in Analytischer Behandlung (Non-Euclidean Space Forms in Analytic Treatment, 1885), where he also pointed out that the embedded torus is contained in a $3$-sphere. The name "Clifford torus" was given by Klein in Zur Nicht-Euklidischen Geometrie (On Non-Euclidean Geometry, 1890), in this book he also used Clifford's identification method to construct the "Klein bottle".

The Clifford torus was one of the first non-trivial examples of "space forms", manifolds locally isometric to a classical geometry (flat, spherical, hyperbolic, etc.). The term "space form" ( "Raumformen" in German) was coined by Killing in 1876-8, in 1891 he called them Clifford-Klein space forms, and showed how to construct them as homogeneous spaces (quotients of classical geometries by discrete group actions). Weyl proved in 1916 that all locally flat space forms are homogeneous spaces.

Volkert gives a very nice account of the history of space forms from 1873 to 1925 with references to the original sources. The watershed of 1925 is when Hopf's Zum Clifford-Kleinschen Raumproblem (On the Clifford-Klein Space Form Problem) came out, where he gave a complete list of flat surfaces and three-dimensional spherical space forms (which eventually led him to the discovery of Hopf fibration). Here is the passage concerning Clifford's flat torus:

"Clifford-Klein space forms entered the history of mathematics in 1873 during a talk which was delivered by W. K. Clifford at the meeting of the British Association for the Advancement of Sciences (Bradford, in September 1873) and via an article he published in June 1873. The title of Clifford’s talk was ‘On a surface of zero curvature and finite extension’, the proceedings of the meeting only provide this title. But we know a bit more about it from F. Klein, who attended Clifford’s talk and who described it on several occasions.

In the context of elliptic geometry - which Clifford conceived in Klein’s way as the geometry of the part of projective space limited by a purely imaginary quartic - Clifford described a closed surface which is locally flat, the today so-called Clifford surface (this name was introduced by Klein). This surface is constructed by using (today so called) Clifford parallels; Bianchi later provided a description by moving a circle along an elliptic straight line in such a way that it is always orthogonal to the straight line. So Clifford’s surface is the analogue of a cylinder; but - since it closed - it is often called a torus.

To show its local flatness, Clifford used a dissection into parallelograms which is defined in a natural way by the two sets of parallels (or generators) contained in the surface. By considering the angles of such a parallelogram one sees that their sum equals four right angles so it is a common flat parallelogram. Clifford concluded: ‘The geometry of this surface is the same as that of a finite parallelogram whose opposite sides are regarded as identical.’ This is a very early occurrence of the identification scheme for the torus!


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