Clifford torus

A stereographic projection of a Clifford torus performing a simple rotation

Topologically a rectangle is the fundamental polygon of a torus, with opposite edges sewn together.

In geometric topology, the Clifford torus is the simplest and most symmetric flat embedding of the Cartesian product of two circles Template:Math and Template:Math (in the same sense that the surface of a cylinder is "flat"). It is named after William Kingdon Clifford. The Clifford Torus is embedded in Template:Math, as opposed to in Template:Math. This is necessary since Template:Math and Template:Math each exists in their own independent embedding space Template:Math and Template:Math, the resulting product space will be Template:Math rather than Template:Math. The historically popular view that the Cartesian product of two circles is an Template:Math 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 Template:Math available to it after the first circle consumes Template:Math and Template:Math.

Stated another way, a torus embedded in Template:Math is an asymmetric reduced-dimension projection of the maximally symmetric Clifford torus embedded in Template:Math. 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 Template:Math and Template:Math each has a radius of Template:Math, their Clifford torus product will fit perfectly within the unit 3-sphere Template:Math, which is a 3-dimensional submanifold of Template:Math. When mathematically convenient, the Clifford torus can be viewed as residing inside the complex coordinate space Template:Math, since Template:Math is topologically equivalent to Template:Math.

The Clifford torus is an example of a square torus, because it is isometric to a square with opposite sides identified. (Some video games, including Asteroids, are played on a square torus; anything that moves off one edge of the screen reappears on the opposite edge with the same orientation.) It is further known as a Euclidean 2-torus (the "2" is its topological dimension); figures drawn on it obey Euclidean geometryTemplate:Clarify 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 Template:Math in Template:Math can be parameterized by an angle coordinate:

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

In another copy of Template:Math, take another copy of the unit circle

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

Then the Clifford torus is

$\frac{1}{\sqrt{2}}{S}^1\times \frac{1}{\sqrt{2}}{S}^1=\{\frac{1}{\sqrt{2}}(\cos \theta ,\sin \theta ,\cos \phi ,\sin \phi )\in {ℝ}^4|0\le \theta <2\pi ,0\le \phi <2\pi \}.$

Since each copy of Template:Math is an embedded submanifold of Template:Math, the Clifford torus is an embedded torus in Template:Math

If Template:Math is given by coordinates Template:Math, then the Clifford torus is given by

$x_1^2+y_1^2=x_2^2+y_2^2=\frac{1}{2}.$

This shows that in Template:Math the Clifford torus is a submanifold of the unit 3-sphere Template:Math.

It is easy to verify that the Clifford torus is a minimal surface in Template:Math.

Alternative derivation using complex numbers

It is also common to consider the Clifford torus as an embedded torus in Template:Math. In two copies of Template:Math, we have the following unit circles (still parametrized by an angle coordinate):

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

and

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

Now the Clifford torus appears as

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

As before, this is an embedded submanifold, in the unit sphere Template:Math in Template:Math.

If Template:Math is given by coordinates Template:Math, then the Clifford torus is given by

${\left|z_1\right|}^2={\left|z_2\right|}^2=\frac{1}{2}.$

In the Clifford torus as defined above, the distance of any point of the Clifford torus to the origin of Template:Math is

$\sqrt{\frac{1}{2}{\left|e^{i\theta }\right|}^2+\frac{1}{2}{\left|e^{i\phi }\right|}^2}=1.$

The set of all points at a distance of 1 from the origin of Template:Math 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 (see Heegaard splitting).

Since O(4) acts on Template:Math 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(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). 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 (i.e., great circles that are maximally separated). 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 Template:Math that are the product of circles of radius Template:Math in one 2-plane Template:Math and radius Template:Math in another 2-plane Template:Math are sometimes also called "Clifford tori".

The same circles may be thought of as having radii that are Template:Math and Template:Math for some angle Template:Math in the range Template:Math (where we include the degenerate cases Template:Math and Template:Math).

The union for Template:Math of all of these tori of form

$T_{\theta }=S(\cos \theta )\times S(\sin \theta )$

(where Template:Math denotes the circle in the plane Template:Math defined by having center Template:Math and radius Template:Math) is the 3-sphere Template:Math. Note that we must include the two degenerate cases Template:Math and Template:Math, each of which corresponds to a great circle of Template:Math, and which together constitute a pair of polar great circles.

This torus Template:Math is readily seen to have area

$\mathrm{area}\left(T_{\theta }\right)=4{\pi }^2\cos \theta \sin \theta =2{\pi }^2\sin 2\theta ,$

so only the torus Template:Math has the maximum possible area of Template:Math. This torus Template:Math is the torus Template:Math that is most commonly called the "Clifford torus" – and it is also the only one of the Template:Math that is a minimal surface in Template:Math.

Still more general definition of Clifford tori in higher dimensions

Any unit sphere Template:Math in an even-dimensional euclidean space Template:Math may be expressed in terms of the complex coordinates as follows:

$S^{2n-1}=\{(z_1,\dots ,z_n)\in {𝐂}^n:|z_1{|}^2+\cdots +|z_n{|}^2=1\}.$

Then, for any non-negative numbers Template:Math such that Template:Math, we may define a generalized Clifford torus as follows:

$T_{r_1,\dots ,r_n}=\{(z_1,\dots ,z_n)\in {𝐂}^n:|z_k|=r_k,1⩽k⩽n\}.$

These generalized Clifford tori are all disjoint from one another. We may once again conclude that the union of each one of these tori Template:Math is the unit Template:Math-sphere Template:Math (where we must again include the degenerate cases where at least one of the radii Template:Math).

Properties

• The Clifford torus is "flat": Every point has a neighborhood that can be flattened out onto a piece of the plane without distortion, unlike the standard torus of revolution.
• The Clifford torus divides the 3-sphere into two congruent solid tori. (In a stereographic projection, the Clifford torus appears as a standard torus of revolution. The fact that it divides the 3-sphere equally means that the interior of the projected torus is equivalent to the exterior.)

Uses in mathematics

In symplectic geometry, the Clifford torus gives an example of an embedded Lagrangian submanifold of Template:Math with the standard symplectic structure. (Of course, any product of embedded circles in Template:Math gives a Lagrangian torus of Template:Math, 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. A proof of this conjecture was published by Simon Brendle in 2013.

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

See also

• Duocylinder
• Hopf fibration
• Clifford parallel and Clifford surface
• William Kingdom Clifford