Clifford torus

When mathematically convenient, the Clifford torus can be viewed as residing inside the complex coordinate space C2, since C2 is topologically equivalent to R4.

It is further known as a Euclidean 2-torus (the "2" is its topological dimension); figures drawn on it obey Euclidean geometry[clarification needed] as if it were flat, whereas the surface of a common "doughnut"-shaped torus is positively curved on the outer rim and negatively curved on the inner.

In fact, the Clifford torus divides this 3-sphere into two congruent solid tori (see Heegaard splitting[2]).

Since O(4) acts on R4 by orthogonal transformations, we can move the "standard" Clifford torus defined above to other equivalent tori via rigid rotations.

The six-dimensional group O(4) acts transitively on the space of all such Clifford tori sitting inside the 3-sphere.

The union for 0 ≤ θ ≤ ⁠π/2⁠ of all of these tori of form (where S(r) denotes the circle in the plane R2 defined by having center (0, 0) and radius r) is the 3-sphere S3.

We may once again conclude that the union of each one of these tori Tr1, ..., rn is the unit (2n − 1)-sphere S2n−1 (where we must again include the degenerate cases where at least one of the radii rk = 0).

[3] Clifford tori and their images under conformal transformations are the global minimizers of the Willmore functional.