Clifford torus


In geometric topology, the Clifford torus is the simplest and most symmetric flat embedding of the cartesian product of two circles S1a and S1b. It is named after William Kingdon Clifford. It resides in R4, as opposed to in R3. To see why R4 is necessary, note that if S1a and S1b each exist in their own independent embedding spaces R2a and R2b, the resulting product space will be R4 rather than R3. The historically popular view that the cartesian product of two circles is an R3 torus in contrast requires the highly asymmetric application of a rotation operator to the second circle, since that circle will only have one independent axis z available to it after the first circle consumes x and y.
Stated another way, a torus embedded in R3 is an asymmetric reduced-dimension projection of the maximally symmetric Clifford torus embedded in R4. The relationship is similar to that of projecting the edges of a cube onto a sheet of paper. Such a projection creates a lower-dimensional image that accurately captures the connectivity of the cube edges, but also requires the arbitrary selection and removal of one of the three fully symmetric and interchangeable axes of the cube.
If S1a and S1b each has a radius of, their Clifford torus product will fit perfectly within the unit 3-sphere S3, which is a 3-dimensional submanifold of R4. When mathematically convenient, the Clifford torus can be viewed as residing inside the complex coordinate space C2, since C2 is topologically equivalent to R4.
The Clifford torus is an example of a square torus, because it is isometric to a square with opposite sides identified. It is further known as a Euclidean 2-torus ; figures drawn on it obey Euclidean geometry 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. Although having a different geometry than the standard embedding of a torus in three-dimensional Euclidean space, the square torus can also be embedded into three-dimensional space, by the Nash embedding theorem; one possible embedding modifies the standard torus by a fractal set of ripples running in two perpendicular directions along the surface.

Formal definition

The unit circle S1 in R2 can be parameterized by an angle coordinate:
In another copy of R2, take another copy of the unit circle
Then the Clifford torus is
Since each copy of S1 is an embedded submanifold of R2, the Clifford torus is an embedded torus in = R4.
If R4 is given by coordinates, then the Clifford torus is given by
This shows that in R4 the Clifford torus is a submanifold of the unit 3-sphere S3.
It is easy to verify that the Clifford torus is a minimal surface in S3.

Alternative derivation using complex numbers

It is also common to consider the Clifford torus as an embedded torus in C2. In two copies of C, we have the following unit circles :
and
Now the Clifford torus appears as
As before, this is an embedded submanifold, in the unit sphere S3 in C2.
If C2 is given by coordinates, then the Clifford torus is given by
In the Clifford torus as defined above, the distance of any point of the Clifford torus to the origin of C2 is
The set of all points at a distance of 1 from the origin of C2 is the unit 3-sphere, and so the Clifford torus sits inside this 3-sphere. In fact, the Clifford torus divides this 3-sphere into two congruent solid tori.
Since O acts on R4 by orthogonal transformations, we can move the "standard" Clifford torus defined above to other equivalent tori via rigid rotations. These are all called "Clifford tori". The six-dimensional group O acts transitively on the space of all such Clifford tori sitting inside the 3-sphere. However, this action has a two-dimensional stabilizer since rotation in the meridional and longitudinal directions of a torus preserves the torus. Hence, there is actually a four-dimensional space of Clifford tori. In fact, there is a one-to-one correspondence between Clifford tori in the unit 3-sphere and pairs of polar great circles. Given a Clifford torus, the associated polar great circles are the core circles of each of the two complementary regions. Conversely, given any pair of polar great circles, the associated Clifford torus is the locus of points of the 3-sphere that are equidistant from the two circles.

More general definition of Clifford tori

The flat tori in the unit 3-sphere S3 that are the product of circles of radius r in one 2-plane R2 and radius in another 2-plane R2 are sometimes also called "Clifford tori".
The same circles may be thought of as having radii that are cos and sin for some angle θ in the range .
The union for of all of these tori of form
is the 3-sphere S3.
This torus Tθ is readily seen to have area
so only the torus Tπ/4 has the maximum possible area of 2π2. This torus Tπ/4 is the torus Tθ that is most commonly called the "Clifford torus" – and it is also the only one of the Tθ that is a minimal surface in S3.

Still more general definition of Clifford tori in higher dimensions

Any unit sphere S2n−1 in an even-dimensional euclidean space may be expressed in terms of the complex coordinates as follows:
Then, for any non-negative numbers r1,..., rn such that r12 +... + rn2 = 1, we may define a generalized Clifford torus as follows:
These generalized Clifford tori are all disjoint from one another. We may once again conclude that the union of each one of these tori Tr1,..., rn is the unit -sphere S2n−1.

Properties

In symplectic geometry, the Clifford torus gives an example of an embedded Lagrangian submanifold of C2 with the standard symplectic structure.
The Lawson conjecture states that every minimally embedded torus in the 3-sphere with the round metric must be a Clifford torus. This conjecture was proved by Simon Brendle in 2012.
Clifford tori and their images under conformal transformations are the global minimizers of the Willmore functional.