A Riemannian approximation of the Heisenberg group is given by declaring the following frame field orthonormal in \mathbb{R}^3:

X_s = \frac{\partial}{\partial x} - y \frac{s}{2} \frac{\partial}{\partial t}, \;\; Y_s = \frac{\partial}{\partial x} - y \frac{s}{2} \frac{\partial}{\partial t} \;\; T_s = \sqrt{1-s} \frac{\partial}{\partial t}

 (note that the T_s has changed from the earlier versions).

There are many ways to approximate horisontal curves in \mathbb{H} by geodesics in the approximations \mathbb{H}_s. Here's one approximation, going from a sphere in Euclidean space to the apple sphere in Heisenberg space, by way of apple sets in the intermediate spaces:

 

 

 

 The general equation for these geodesics is:

\gamma_s(t) = \left[\begin{matrix}\sqrt{\left(\frac{r}{c}\right)^2 - \frac{1-s}{s^2}}(\cos(c t)-1)\\ \sqrt{\left(\frac{r}{c}\right)^2 - \frac{1-s}{s^2}}\sin ct\\ \frac{1-s}{s}ct + \frac{s}{2}\left( \left( \frac{r}{c}\right)^2 - \frac{1-s}{s^2}\right) (ct - \sin ct)\end{matrix}\right]

Unfortunately, this is not defined for (rsc)^2<1-s.

Here is another approximation, the first one producing apple sets. In this animation the sphere is approximated on the Euclidean end by a cylinder of height 2\pi, and the bubble set is a bit wrong in the animation.


The following are the corrected versions of the above animations, though with modified curvature bounds, so that the curvature parameter changes with s changing. Here, apple spheres start off as infinite cylinders in Euclidean space.

 

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Last Updated ( Tuesday, 23 June 2009 12:10 )