Quantum Markov chain

A stereographic projection of a Clifford torus performing a simple rotation

Topologically a square is the fundamental polygon of a torus, with the left and right edges sewn together, and the top and bottom edges sewn together.

In geometric topology, the Clifford torus is a special kind of torus sitting inside R⁴. Alternatively, it can be seen as a torus sitting inside C² since C² is topologically the same space as R⁴. Furthermore, every point of the Clifford torus lies at a fixed distance from the origin; therefore, it can also be viewed as sitting inside a 3-sphere.

The Clifford torus is also known as a square torus, because it is isometric to a square with side length 2π and with opposite sides identified.

Formal definition

The unit circle S¹ in R² can be parameterized by an angle coordinate:

$S^{1}=\{(\cos {\theta },\sin {\theta })\,|\,0\leq \theta <2\pi \}.$

In another copy of R², take another copy of the unit circle

$S^{1}=\{(\cos {\phi },\sin {\phi })\,|\,0\leq \phi <2\pi \}.$

Then the Clifford torus is

$S^{1}\times S^{1}=\{(\cos {\theta },\sin {\theta },\cos {\phi },\sin {\phi })\,|\,0\leq \theta <2\pi ,0\leq \phi <2\pi \}.$

Since each copy of S¹ is an embedded submanifold of R², the Clifford torus is an embedded torus in R² × R² = R⁴.

If R⁴ is given by coordinates (x₁, y₁, x₂, y₂), then the Clifford torus is given by

$x_{1}^{2}+y_{1}^{2}=1=x_{2}^{2}+y_{2}^{2}.\,$

Alternate definitions

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 (still parametrized by an angle coordinate):

$S^{1}=\{e^{i\theta }\,|\,0\leq \theta <2\pi \}$

and

$S^{1}=\{e^{i\phi }\,|\,0\leq \phi <2\pi \}.$

Now the Clifford torus appears as

$S^{1}\times S^{1}=\{(e^{i\theta },e^{i\phi })\,|\,0\leq \theta <2\pi ,0\leq \phi <2\pi \}.$

As before, this is an embedded submanifold, in this case of C².

If C² is given by coordinates (z₁, z₂), then the Clifford torus is given by

$\left|z_{1}\right|^{2}=1=\left|z_{2}\right|^{2}.$

In the Clifford torus as defined above, the distance of any point of the Clifford torus to the origin of C² is

${\sqrt {\left|e^{i\theta }\right|^{2}+\left|e^{i\phi }\right|^{2}}}={\sqrt {2}}.$

The set of all points at a distance of √2 from the origin of C² is a 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. (See Heegaard splitting.)

Instead of defining the Clifford torus as the product of two unit circles, it is also common to use two circles of radius 1/√2. (For example, Paul Norbury uses this convention when describing the Lawson Conjecture.) With the alternate radius of 1/√2, the Clifford torus instead sits in the unit 3-sphere S³.

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. However, this action has a two-dimensional stabilizer (see group action) since rotation in the meridional and longitudinal directions of a torus preserves the torus (as opposed to moving it to a different torus). So there is actually a four-dimensional space of Clifford tori.

Uses in mathematics

In symplectic geometry, the Clifford torus gives an example of an embedded Lagrangian submanifold of C² with the standard symplectic structure. (Of course, any product of embedded circles in C gives a Lagrangian torus of C², so these need not be Clifford tori.)

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.

See also

• Duocylinder
• Hopf fibration