Given a surface in Heisenberg space , the tangent space to 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 , there is a single line in that is horisontal, and one may ask for a unit-speed horisontal line through . 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:

 

 

 

 

A torus sitting along the XY plane does not have any critical points, and the horisontal lines are either dense or close up, depending on the rationality of a parameter associated to the torus (see the next post on the Hopf fibration).

 

 

 

Aligning the torus along the XZ plane displays the line of characteristic points: the numerically found horisontal curves cannot extend uniquely past these. You can also see that the saddle points of the torus are critical points:

 

 

Spherical Coordinates

Several proofs related to the isoperimetric inequality in {tex}\mathbb R ^n {\tex} use spherical coordinates. One can think of these as specifying a cone (the {tex}\phi{\tex} parameter), a geodesic on that cone ({tex}\theta{\tex}) and a location on that geodesic ({tex}r{\tex}).

In Heisenberg space, paraboloids are the natural scaling-invariant analogue of cones. In the XY plane, the geodesics are just radial lines through the origin. For other parabolas, we see spirals. In the following video, we increase the excentricity of the parabola and watch the geodesics change:

 

Here's a view from below:

 For some more horisontal lines, see the next post on the Hopf fibration.

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Last Updated ( Friday, 08 May 2009 01:53 )