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Monday, 11 May 2009 21:43 | Author: Anton

Riemannian Approximation

Thursday, 14 May 2009 22:51 | Author: Anton

The fields on the Heisenberg group $\mathbb H$ that gives it its sub-Riemannian and metric structure is given by

$X = \frac{\partial}{\partial x} - \frac{1}{2}y \frac{\partial}{\partial t}, Y = \frac{\partial}{\partial y} + \frac{1}{2}x \frac{\partial}{\partial t}$

These are declared orthonormal, and only lines whose velocities are given by $aX + bY$ have a (finite) speed. One may view this setup as a degenerate Riemannian structure, where a third vector field (say, $s \frac{\partial}{\partial t}$ ) has degenerated as s was sent to zero.

The Riemannian approximation I find natural interpolates between Euclidean and Heisenberg geometries, and has not yet been explored. Here are the frame fields:

$X_s = \frac{\partial}{\partial x} - \frac{s}{2} y \frac{\partial}{\partial t}, Y_s = \frac{\partial}{\partial y} + \frac{s}{2} x \frac{\partial}{\partial t}, T_s = (1-s) \frac{\partial}{\partial t}$

At s=0, this is the standard Euclidean frame field, and at s=1 it degenerates to the Heisenberg structure. In between, it gives Riemannian structures on $\mathbb R^3$ with properties that degenerate as s goes to 1.

As with any Riemannian structure, geodesics through a given point with assigned direction exist and are unique. They can be found by solving the geodesic equation

$\nabla_{\gamma\prime} \gamma\prime = 0$

for the Levi-Civita connection corresponding to the orthonormal frame field. Given a curve $\gamma = (x,y,z)$ and some calculations, the condition on $\gamma$ becomes:

$(1-s)^2 x\prime\prime + s y\prime ( z\prime + \frac{s}{2}( x\prime y - y\prime x) )=0$

$(1-s)^2 y\prime \prime - s x\prime ( z\prime + \frac{s}{2}( x\prime y - y\prime x) )=0$

$( z\prime + \frac{s}{2}( x\prime y - y\prime x) )\prime =0$

These equations are easily solved. The following animations show, as s changes, geodesics leaving the origin with fixed initial velocity. The length of the geodesics changes with s since otherwise they quickly become too small to see.

Last Updated ( Friday, 22 May 2009 11:36 )

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