In geometric topology, the Clifford torus is the simplest and most symmetric flat embedding of the Cartesian product of two circles and (in the same sense that the surface of a cylinder is "flat"). It is named after William Kingdon Clifford. It resides in, as opposed to in . To see why is necessary, note that if and each exists in its own independent embedding space and, the resulting product space will be rather than . The historically popular view that the Cartesian product of two circles is an 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 available to it after the first circle consumes and .
Stated another way, a torus embedded in is an asymmetric reduced-dimension projection of the maximally symmetric Clifford torus embedded in . 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 and each has a radius of, their Clifford torus product will fit perfectly within the unit 3-sphere, which is a 3-dimensional submanifold of . When mathematically convenient, the Clifford torus can be viewed as residing inside the complex coordinate space, since is topologically equivalent to .
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 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]
The unit circle in can be parameterized by an angle coordinate:
S1=l\{(\cos\theta,\sin\theta)|0\leq\theta<2\pir\}.
In another copy of, take another copy of the unit circle
S1=l\{(\cos\varphi,\sin\varphi)|0\leq\varphi<2\pir\}.
\tfrac{1}{\sqrt{2}}S1 x \tfrac{1}{\sqrt{2}}S1=\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 is an embedded submanifold of, the Clifford torus is an embedded torus in
If is given by coordinates, then the Clifford torus is given by
| 2 | |
| x | |
| 1 |
+
| 2 | |
| y | |
| 1 |
=
| 2 | |
| x | |
| 2 |
+
| 2 | |
| y | |
| 2 |
=\tfrac{1}{2}.
This shows that in the Clifford torus is a submanifold of the unit 3-sphere .
It is easy to verify that the Clifford torus is a minimal surface in .
It is also common to consider the Clifford torus as an embedded torus in . In two copies of, we have the following unit circles (still parametrized by an angle coordinate):
S1=\left\{\left.ei\theta\right|0\leq\theta<2\pi\right\}
S1=\left\{\left.ei\varphi\right|0\leq\varphi<2\pi\right\}.
\tfrac{1}{\sqrt{2}}S1 x \tfrac{1}{\sqrt{2}}S1=\left\{\left.\tfrac{1}{\sqrt{2}}\left(ei\theta,ei\varphi\right)\right|0\leq\theta<2\pi,0\leq\varphi<2\pi\right\}.
If is given by coordinates, then the Clifford torus is given by
\left|z1\right|2=\left|z2\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 is
\sqrt{\tfrac{1}{2}\left|ei\theta\right|2+\tfrac{1}{2}\left|ei\varphi\right|2}=1.
Since O(4) acts on 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.
The flat tori in the unit 3-sphere that are the product of circles of radius in one 2-plane and radius in another 2-plane are sometimes also called "Clifford tori".
The same circles may be thought of as having radii that are and for some angle in the range (where we include the degenerate cases and).
The union for of all of these tori of form
T\theta=S(\cos\theta) x S(\sin\theta)
(where denotes the circle in the plane defined by having center and radius) is the 3-sphere . Note that we must include the two degenerate cases and, each of which corresponds to a great circle of, and which together constitute a pair of polar great circles.
This torus is readily seen to have area
\operatorname{area}\left(T\theta\right)=4\pi2\cos\theta\sin\theta=2\pi2\sin2\theta,
so only the torus has the maximum possible area of . This torus is the torus that is most commonly called the "Clifford torus" – and it is also the only one of the that is a minimal surface in .
Any unit sphere in an even-dimensional euclidean space may be expressed in terms of the complex coordinates as follows:
S2n-1=\left\{(z1,\ldots,zn)\inCn:
| 2 | |
| |z | |
| 1| |
+ … +
| 2 | |
| |z | |
| n| |
=1\right\}.
Then, for any non-negative numbers such that, we may define a generalized Clifford torus as follows:
| T | |
| r1,\ldots,rn |
=l\{(z1,\ldots,zn)\inCn:|zk|=rk,~1\leqslantk\leqslantnr\}.
These generalized Clifford tori are all disjoint from one another. We may once again conclude that the union of each one of these tori is the unit -sphere (where we must again include the degenerate cases where at least one of the radii).
In symplectic geometry, the Clifford torus gives an example of an embedded Lagrangian submanifold of with the standard symplectic structure. (Of course, any product of embedded circles in gives a Lagrangian torus of, 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.[2]
Clifford tori and their images under conformal transformations are the global minimizers of the Willmore functional.