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):

 

Distance-minimizing geodesics between two points are again usually unique, except for the case of "bubble sets".This is what bubble sets look like in S^3 (after stereographic projection; the animation varies the curvature of the geodesics):
 

 
Another way to interpret these sets is as the collection of geodesics of a given curvature leaving a given point. Notice how one of the cluster points is stationary, while the "destination" point moves closer to it as the curvature increases (from 0 to 1).
 
Given the more immediate connection with complex hyperbolic plane \mathbb{H}^2_{\mathbb{C}} (S^3 is its boundary and the distribution is Isom(\mathbb{H}^2_{\mathbb{C}})-invariant), one would expect to see a connection between natural objects in S^3 and those appearing in \partial\mathbb{H}^2_{\mathbb{C}}.
 
The main surface arising from hyperbolic space is the "spinal sphere", which I'll describe in more detail later. Given two points in \partial\mathbb{H}^2_{\mathbb{C}}, there is a unique spinal sphere defined by them. In the case when the spinal sphere is "vertical", it corresponds with a Cygan sphere, and the two points are its north and south pole.
 
Unfortunately, the bubble set with its two special points is not (at least in general) the same as the spinal sphere defined by those points:
What you're seeing in the picture is one of the bubble sets from the video (the smaller object), the spinal sphere defined by its two focus points (the larger object), and the spinal sphere's "spine" (the circle). Unfortunately, these aren't the same object. At least not for this metric on the sphere.
 
Note: the bubble set was drawn using the parametrization in the Hurtado-Rosales 2006 paper "Area-Stationary Surfaces Inside the Sub-Riemannian Three-Sphere".

 

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Last Updated ( Thursday, 14 May 2009 20:14 )