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 S¹
_a and S¹
_b (in the same sense that the surface of a cylinder is "flat"). 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 S¹
_a and S¹
_b each exists in its own independent embedding space R²
_a and R²
_b, 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 S¹
_a and S¹
_b each has a radius of ⁠1/√2⁠, 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. (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 geometry ^{[ clarification needed ]} 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. ^[1]

Formal definition

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

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

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

${\displaystyle S^{1}={\bigl \{}(\cos \varphi ,\sin \varphi )\,{\big |}\,0\leq \varphi <2\pi {\bigr \}}.}$

Then the Clifford torus is

${\displaystyle {\tfrac {1}{\sqrt {2}}}S^{1}\times {\tfrac {1}{\sqrt {2}}}S^{1}=\left\{\left.{\tfrac {1}{\sqrt {2}}}(\cos \theta ,\sin \theta ,\cos \varphi ,\sin \varphi )\,\right|\,0\leq \theta <2\pi ,0\leq \varphi <2\pi \right\}.}$

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

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

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 (still parametrized by an angle coordinate):

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

and

${\displaystyle S^{1}=\left\{\left.e^{i\varphi }\,\right|\,0\leq \varphi <2\pi \right\}.}$

Now the Clifford torus appears as

${\displaystyle {\tfrac {1}{\sqrt {2}}}S^{1}\times {\tfrac {1}{\sqrt {2}}}S^{1}=\left\{\left.{\tfrac {1}{\sqrt {2}}}\left(e^{i\theta },e^{i\varphi }\right)\,\right|\,0\leq \theta <2\pi ,0\leq \varphi <2\pi \right\}.}$

As before, this is an embedded submanifold, in the unit sphere S³ in C².

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

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

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

${\displaystyle {\sqrt {{\tfrac {1}{2}}\left|e^{i\theta }\right|^{2}+{\tfrac {1}{2}}\left|e^{i\varphi }\right|^{2}}}=1.}$

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 (see Heegaard splitting ^[2] ).

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. 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. ^[2] 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 S³ that are the product of circles of radius r in one 2-plane R² and radius √1 − r² in another 2-plane R² are sometimes also called "Clifford tori".

The same circles may be thought of as having radii that are cos θ and sin θ for some angle θ in the range 0 ≤ θ ≤ ⁠π/2⁠ (where we include the degenerate cases θ = 0 and θ = ⁠π/2⁠).

The union for 0 ≤ θ ≤ ⁠π/2⁠ of all of these tori of form

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

(where S(r) denotes the circle in the plane R² defined by having center (0, 0) and radius r) is the 3-sphere S³. Note that we must include the two degenerate cases θ = 0 and θ = ⁠π/2⁠, each of which corresponds to a great circle of S³, and which together constitute a pair of polar great circles.

This torus T_θ is readily seen to have area

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

so only the torus T_⁠π/4⁠ has the maximum possible area of 2π². This torus T_⁠π/4⁠ is the torus T_θ that is most commonly called the "Clifford torus" – and it is also the only one of the T_θ 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 R²ⁿ = Cⁿ may be expressed in terms of the complex coordinates as follows:

${\displaystyle S^{2n-1}=\left\{(z_{1},\ldots ,z_{n})\in \mathbf {C} ^{n}:|z_{1}|^{2}+\cdots +|z_{n}|^{2}=1\right\}.}$

Then, for any non-negative numbers r₁, ..., r_n such that r₁² + ... + r_n² = 1, we may define a generalized Clifford torus as follows:

${\displaystyle T_{r_{1},\ldots ,r_{n}}={\bigl \{}(z_{1},\ldots ,z_{n})\in \mathbf {C} ^{n}:|z_{k}|=r_{k},~1\leqslant k\leqslant n{\bigr \}}.}$

These generalized Clifford tori are all disjoint from one another. We may once again conclude that the union of each one of these tori T_{r1, ..., rn} is the unit (2n − 1)-sphere S²ⁿ⁻¹ (where we must again include the degenerate cases where at least one of the radii r_k = 0).

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 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. A proof of this conjecture was published by Simon Brendle in 2013. ^[3]

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

References

1. Borrelli, V.; Jabrane, S.; Lazarus, F.; Thibert, B. (April 2012), "Flat tori in three-dimensional space and convex integration", Proceedings of the National Academy of Sciences, 109 (19): 7218–7223, doi: 10.1073/pnas.1118478109 , PMC   3358891 , PMID   22523238 .
2. Norbs, P. (September 2005), "The 12th problem" (PDF), The Australian Mathematical Society Gazette, 32 (4): 244–246
3. Brendle, Simon (2013), "Embedded minimal tori in S³ and the Lawson conjecture", Acta Mathematica, 211 (2): 177–190, arXiv: 1203.6597 , doi:10.1007/s11511-013-0101-2 ; see reviews by João Lucas Marques Barbosa (MR 3143888) and Ye-Lin Ou ( Zbl   1305.53061)