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The Heisenberg 3-Sphere

Math - Heisenberg09
$S^3$ has a Lie group structure as a subset of the quaternions. Given a point $p \in S^3$ , a choice of a two-dimensional subspace of $T_p S^3$ can be extended to a 2-dimensional distribution on the entire manifold. With an appropriate choice of stereographic projection, this corresponds to the sub-Riemannian distribution on the Heisenberg group. The restriction of the Euclidean inner product to $T_pS^3$ also extends to the other points, giving a sub-Riemannian structure and Carnot-Caratheodory (CC) metric on $S^3$ . As with the Heisenberg group, one might consider the metric properties of this space. What are the geodesics? Are they unique? What do the spheres look like? The geodesics, it turns out come in two flavors: homeomorphic to $\mathbb R$ and $S^1$ . As with the Heisenberg group, there is a curvature $\lambda$ to a geodesic, and if $\frac{\lambda}{1-\lambda}$ is irrational, the geodesic is dense in a Clifford torus (see post on the Hopf Fibration):
Last Updated on Thursday, 14 May 2009 20:14
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Mathematica Notebook: Horisontal, Hopf, Lens

Math - Heisenberg09
The images and animations in the previous four posts were made using Mathematica. Here is the notebook Horisontal Curves, Hopf Fibration, and Lens Spaces that contains commands for producing most of the images and animations I posted. Feel free to contact me with any questions or comments you have about it.
Last Updated on Wednesday, 13 May 2009 11:13

Math - Heisenberg09
The Lens space $L(p,q)$ is the quotient of $S^3$ by an action of $\mathbb{Z}_p$ . The group acts on $\mathbb{C}^2$ by the $p^{th}$ root of unity in the first coordinate and the $q^{th}$ power of that in the second coordinate, and the action restricts to $S^3$ . So why a "Lens" space, and how does one visualize it? I drew $L(5,2)$ by drawing a lattice in $S^3$ invariant under the group action. Here's the picture (after stereographic projection):
Last Updated on Tuesday, 12 May 2009 11:45
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The Hopf Fibration

Math - Heisenberg09
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 on Thursday, 07 May 2009 16:47
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Horizontal Lines and Spherical Coordinates

Math - Heisenberg09
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 on Thursday, 07 May 2009 19:53
Read more... [Horizontal Lines and Spherical Coordinates]

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Heisenberg Space

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