Clifford Torus - Alternate Definitions

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):

and

Now the Clifford torus appears as

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

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 √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.