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 differential geometry, the Clifford torus is the standard embedding of the 2-torus as a product of circles in Euclidean space Template:Math (equivalently Template:Math). For radii Template:Math it can be written as Template:Math, or in complex coordinates as the set of Template:Math with $|z_1|=a$ and $|z_2|=b$. With the induced metric it is a flat torus (its Gaussian curvature vanishes), isometric to a rectangular torus; when Template:Math it is a square torus.

If Template:Math, then Template:Math lies in the unit 3-sphere Template:Math. The case Template:Math is a minimal surface in Template:Math and is often called the minimal Clifford torus; its images under the isometries of Template:Math are also minimal. The Clifford torus is named after William Kingdon Clifford and is closely related to Clifford parallelism and the Hopf fibration.

A flat 2-torus admits no Template:Math (in particular, no smooth) isometric embedding into Template:Math, but by the Nash–Kuiper theorem it does admit Template:Math isometric embeddings into Template:Math (constructed via convex integration).

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.

Alternative embeddings

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.

See also

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