Clifford torus

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 $S 1 a$ and $S 1 b$ (in the same sense that the surface of a cylinder is "flat"). It is named after William Kingdon Clifford. It resides in $R 4$ , as opposed to in $R 3$ . To see why $R 4$ is necessary, note that if $S 1 a$ and $S 1 b$ each exists in its own independent embedding space $R 2 a$ and $R 2 b$ , the resulting product space will be $R 4$ rather than $R 3$ . The historically popular view that the Cartesian product of two circles is an $R 3$ 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 $R 3$ is an asymmetric reduced-dimension projection of the maximally symmetric Clifford torus embedded in $R 4$ . 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.

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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 $S 1$ in $R 2$ can be parameterized by an angle coordinate:

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

In another copy of $R 2$ , take another copy of the unit circle

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

Then the Clifford torus is

{\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 $S 1$ is an embedded submanifold of $R 2$ , the Clifford torus is an embedded torus in $R 2 \times R 2 = R 4 .$

If $R 4$ is given by coordinates $(x 1, y 1, x 2, y 2)$ , then the Clifford torus is given by

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

This shows that in $R 4$ the Clifford torus is a submanifold of the unit 3-sphere $S 3$ .

It is easy to verify that the Clifford torus is a minimal surface in $S 3$ .

Alternative derivation using complex numbers

It is also common to consider the Clifford torus as an embedded torus in $C 2$ . In two copies of $C$ , we have the following unit circles (still parametrized by an angle coordinate):

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

and

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

Now the Clifford torus appears as

{\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 $S 3$ in $C 2$ .

If $C 2$ is given by coordinates $(z 1, z 2)$ , then the Clifford torus is given by

\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 $C 2$ is

{\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 $C 2$ 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 $R 4$ 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 $S 3$ that are the product of circles of radius $r$ in one 2-plane $R 2$ and radius $\sqrt 1 - r 2$ in another 2-plane $R 2$ 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 $0 \leq θ \leq ⁠ π / 2 ⁠$ (where we include the degenerate cases $θ = 0$ and $θ = ⁠ π / 2 ⁠$ ).

The union for $0 \leq θ \leq ⁠ π / 2 ⁠$ of all of these tori of form

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

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

This torus $T θ$ is readily seen to have area

\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 $2 π 2$ . 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 $S 3$ .

Still more general definition of Clifford tori in higher dimensions

Any unit sphere $S 2 n -1$ in an even-dimensional euclidean space $R 2 n = C n$ may be expressed in terms of the complex coordinates as follows:

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 $r 1, ..., r n$ such that $r 12 + ... + r n 2 = 1$ , we may define a generalized Clifford torus as follows:

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 $T r 1, ..., r n$ is the unit $(2 n - 1)$ -sphere $S 2 n -1$ (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 $C 2$ with the standard symplectic structure. (Of course, any product of embedded circles in $C$ gives a Lagrangian torus of $C 2$ , 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.

References

^ 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.
^ ^a ^b Norbs, P. (September 2005), "The 12th problem" (PDF), The Australian Mathematical Society Gazette, 32 (4): 244–246
^ Brendle, Simon (2013), "Embedded minimal tori in $S 3$ 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 (MR3143888) and Ye-Lin Ou (Zbl 1305.53061)

[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.

[Norbs-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 $S 3$ 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 (MR3143888) and Ye-Lin Ou (Zbl 1305.53061)

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