In differential geometry, the Clifford torus is the standard embedding of the 2-torus as a product of circles in Euclidean space (equivalently ). For radii it can be written as , or in complex coordinates as the set of with and . With the induced metric it is a flat torus (its Gaussian curvature vanishes), isometric to a rectangular torus; when it is a square torus.
If , then lies in the unit 3-sphere . The case is a minimal surface in and is often called the minimal Clifford torus; its images under the isometries of 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 (in particular, no smooth) isometric embedding into , but by the NashâÂÂKuiper theorem it does admit isometric embeddings into (constructed via convex integration).
The unit circle in can be parameterized by an angle coordinate:
In another copy of , take another copy of the unit circle
Then the Clifford torus is
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
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):
and
Now the Clifford torus appears as
As before, this is an embedded submanifold, in the unit sphere in .
If is given by coordinates , then the Clifford torus is given by
In the Clifford torus as defined above, the distance of any point of the Clifford torus to the origin of is
The set of all points at a distance of 1 from the origin of 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 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
(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
so only the torus has the maximum possible area of . This torus is the torus that is most commonly called the "Clifford torus" â 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:
Then, for any non-negative numbers such that , we may define a generalized Clifford torus as follows:
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.
Clifford tori and their images under conformal transformations are the global minimizers of the Willmore functional.
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.