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The Hopf Fibration

Thursday, 07 May 2009 16:13 | Author: Anton

The map $\mathbb{C}^2 \rightarrow \mathbb{CP}^1 = S^2$ restricted to $S^3 \subset \mathbb C^2$ is called the Hopf map and is usually viewed as a map (fibration) $S^3 \rightarrow S^2$ whose fibers (preimage of a point) are circles.

The preimage of a circle in $S^2$ is called a Clifford torus. These fill out all of $S^3$ except for two linked circles, and are in turn foliated by circles.

To visualize the Hopf fibration, we apply stereographic projection $S^3 \rightarrow \mathbb{R}^3$ or, better yet, into the Heisenberg group. Here's a picture of three Clifford tori with Heisenberg horisontal lines:

$Hopf Fibration with Heisenberg Horisontal Lines$

Last Updated ( Thursday, 07 May 2009 16:47 )

Horizontal Lines and Spherical Coordinates

Math - Heisenberg09

Thursday, 07 May 2009 13:17 | Author: Anton

Given a surface $S$ in Heisenberg space $\mathbb{H}$ , the tangent space to $S$ at a given point is generally not the same as the horisontal plane. In fact, the set of points where the two are equal, called the critical points, at worst form a collection of lines on the surface.

At every non-critical point $p$ , there is a single line in $T_p S$ that is horisontal, and one may ask for a unit-speed horisontal line through $p$ . Given a parametrization of the surface, this becomes a system of PDE's that Mathematica can solve using NDSolve.

In the case of the Euclidean unit sphere at the origin, there are two critical points: the north and south pole. The rest of the sphere is foliated by horisontal lines:

$Horisontal Lines on Euclidean Sphere$

Last Updated ( Thursday, 07 May 2009 19:53 )

Heisenberg Space

Math - Heisenberg09

Thursday, 07 May 2009 12:40 | Author: Anton

The Heisenberg group $\mathbb {H}_i$ is $\mathbb{R}^3$ with the non-commutative group law $(x_1,y_1,t_1)+(x_2,y_2,t_2) = \left(x_1+x_2, y_1+y_2, t_1+t_2+\frac{1}{2}( x_1 y_2 - y_1 x_2) \right)$ .
This group law commutes with the dilation $(x,y,t) \mapsto (r x, ry, r^2 t)$ , so $\mathbb{H}_i$ cannot have a Riemannian metric on it. Instead, it is equipped with a sub-Riemannian metric: an inner product on a distribution over $\mathbb{H}_i$ . At the origin, this distribution gives the XY plane, and the inner product is just the Euclidean inner product on this two-dimensional subspace. This horizontal distribution and inner product is defined at the other points using invariance under left multiplication.
Here is the standard picture of the horizontal distribution. We just show what happens along the XY plane. Vertical translation by $(0,0,t)$ acts as the identity on the distribution.

$Distribution on Heisenberg Space, generic view$

Last Updated ( Friday, 08 May 2009 14:10 )