Clifford torus

In differential geometry, the Clifford torus is the standard embedding of the 2-torus as a product of circles in Euclidean space R⁴ (equivalently C²). For radii a,b>0 it can be written as S¹(a) × S¹(b) ⊂ R² × R² ≅R⁴, or in complex coordinates as the set of (z₁,z₂) with ${\displaystyle |z_{1}|=a}$ and ${\displaystyle |z_{2}|=b}$. With the induced metric it is a flat torus (its Gaussian curvature vanishes), isometric to a rectangular torus; when a=b it is a square torus.

If a²+b²=1, then T_a,b lies in the unit 3-sphere S³ ⊂ R⁴. The case a=b=1/√2 is a minimal surface in S³ and is often called the minimal Clifford torus; its images under the isometries of S³ are also minimal. The Clifford torus is named after William Kingdon Clifford and is closely related to Clifford parallelism and the Hopf fibration.

A flat 2-torus admits no C² (in particular, no smooth) isometric embedding into R³, but by the Nash–Kuiper theorem it does admit C¹ isometric embeddings into R³ (constructed via convex integration).