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| [[File:Clifford-torus.gif|thumb|right|256px|A [[stereographic projection]] of a Clifford torus performing a [[SO(4)#Geometry_of_4D_rotations|simple rotation]]]]
| | Claude is her name and she completely digs that name. Interviewing is what I do in my working day occupation. The factor she adores most is to play handball but she can't make it her occupation. Some time in the past he chose to live in Idaho.<br><br>Here is my page :: [http://Glskating.com/groups/great-ideas-about-auto-repair-that-you-can-use/ http://Glskating.com/groups/great-ideas-about-auto-repair-that-you-can-use] |
| [[File:TorusAsSquare.svg|thumb|Topologically a [[square]] is the [[fundamental polygon]] of a torus, with the left and right edges sewn together, and the top and bottom edges sewn together.]]
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| In [[geometric topology]], the '''Clifford torus''' is a special kind of [[torus]] sitting inside '''R'''<sup>4</sup>. Alternatively, it can be seen as a torus sitting inside '''C'''<sup>2</sup> since '''C'''<sup>2</sup> is topologically the same space as '''R'''<sup>4</sup>. Furthermore, every point of the Clifford torus lies at a fixed distance from the origin; therefore, it can also be viewed as sitting inside a [[3-sphere]].
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| The Clifford torus is also known as a '''square torus''', because it is [[Isometry|isometric]] to a [[Square (geometry)|square]] with side length 2π and with opposite sides identified. | |
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| == Formal definition ==
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| The [[unit circle]] S<sup>1</sup> in '''R'''<sup>2</sup> can be parameterized by an angle coordinate:
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| : <math>S^1 = \{ ( \cos{\theta}, \sin{\theta} ) \, | \, 0 \leq \theta < 2\pi \}.</math>
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| In another copy of '''R'''<sup>2</sup>, take another copy of the unit circle
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| : <math>S^1 = \{ ( \cos{\phi}, \sin{\phi} ) \, | \, 0 \leq \phi < 2\pi \}.</math>
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| Then the Clifford torus is
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| : <math>S^1 \times S^1 = \{ ( \cos{\theta}, \sin{\theta}, \cos{\phi}, \sin{\phi} ) \, | \, 0 \leq \theta < 2\pi, 0 \leq \phi < 2\pi \}.</math>
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| Since each copy of S<sup>1</sup> is an embedded [[submanifold]] of '''R'''<sup>2</sup>, the Clifford torus is an embedded torus in '''R'''<sup>2</sup> × '''R'''<sup>2</sup> = '''R'''<sup>4</sup>.
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| If '''R'''<sup>4</sup> is given by coordinates (''x''<sub>1</sub>, ''y''<sub>1</sub>, ''x''<sub>2</sub>, ''y''<sub>2</sub>), then the Clifford torus is given by
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| : <math>x_1^2 + y_1^2 = 1 = x_2^2 + y_2^2. \, </math>
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| == Alternate definitions ==
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| It is also common to consider the Clifford torus as an [[embedding|embedded]] torus in '''C'''<sup>2</sup>. In two copies of '''C''', we have the following unit circles (still parametrized by an angle coordinate):
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| : <math>S^1 = \{ e^{i\theta} \, | \, 0 \leq \theta < 2\pi \}</math>
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| and
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| : <math>S^1 = \{ e^{i\phi} \, | \, 0 \leq \phi < 2\pi \}.</math> | |
| Now the Clifford torus appears as
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| : <math>S^1 \times S^1 = \{ ( e^{i\theta}, e^{i\phi} ) \, | \, 0 \leq \theta < 2\pi, 0 \leq \phi < 2\pi \}.</math> | |
| As before, this is an embedded submanifold, in this case of '''C'''<sup>2</sup>.
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| If '''C'''<sup>2</sup> is given by coordinates (''z''<sub>1</sub>, ''z''<sub>2</sub>), then the Clifford torus is given by
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| : <math>\left| z_{1} \right|^{2} = 1 = \left| z_{2} \right|^{2}.</math> | |
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| In the Clifford torus as defined above, the distance of any point of the Clifford torus to the origin of '''C'''<sup>2</sup> is
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| : <math>\sqrt{ \left| e^{i\theta} \right|^2 + \left| e^{i\phi} \right|^2 } = \sqrt{2}.</math>
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| The set of all points at a distance of √2 from the origin of '''C'''<sup>2</sup> is a 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 torus|solid tori]]. (See [[Heegaard splitting]].)
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| Instead of defining the Clifford torus as the product of two unit circles, it is also common to use two circles of radius 1/√2. (For example, Paul Norbury uses this convention when describing the [[Hsiang–Lawson's conjecture|Lawson Conjecture]].<ref name="Norbury">
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| {{cite journal
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| | last = Norbury
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| | first = Paul
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| | authorlink =
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| | coauthors =
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| | title = The 12th problem
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| | journal = The Australian Mathematical Society Gazette
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| | volume = 32
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| | issue = 4
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| | pages = 244–246
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| | publisher =
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| | date = September, 2005
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| | url = http://www.austms.org.au/Publ/Gazette/2005/Sep05/millennium.pdf
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| | format = [[PDF]]
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| | doi =
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| | id =
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| | accessdate = }}</ref>) With the alternate radius of 1/√2, the Clifford torus instead sits in the unit 3-sphere S<sup>3</sup>.
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| Since [[Orthogonal group|''O''(4)]] acts on '''R'''<sup>4</sup> by [[orthogonal transformation]]s, we can move the "standard" Clifford torus defined above to other equivalent tori via rigid rotations. 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). So there is actually a four-dimensional space of Clifford tori.<ref name="Norbury" />
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| == Uses in mathematics ==
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| In [[symplectic geometry]], the Clifford torus gives an example of an embedded [[Lagrangian submanifold]] of '''C'''<sup>2</sup> with the standard symplectic structure. (Of course, any product of embedded circles in '''C''' gives a Lagrangian torus of '''C'''<sup>2</sup>, so these need not be Clifford tori.)
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| The [[Hsiang–Lawson's conjecture|Lawson conjecture]] states that every [[minimal surface|minimally embedded]] torus in the 3-sphere with the [[Metric tensor#The round metric on a sphere|round metric]] must be a Clifford torus. This conjecture was proved by Simon Brendle in 2012.
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| == See also ==
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| * [[Duocylinder]]
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| * [[Hopf fibration]]
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| == References ==
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| <references/>
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| [[Category:Geometric topology]]
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Claude is her name and she completely digs that name. Interviewing is what I do in my working day occupation. The factor she adores most is to play handball but she can't make it her occupation. Some time in the past he chose to live in Idaho.
Here is my page :: http://Glskating.com/groups/great-ideas-about-auto-repair-that-you-can-use