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Heisenberg Space

Math - Heisenberg09
Written by Anton
Thursday, 07 May 2009 18:40
The Heisenberg group $\mathbb {H}_i$ is $\mathbb{R}^3$ with the non-commutative group law $(x_1,y_1,t_1)+(x_2,y_2,t_2) = \left(x_1+x_2, y_1+y_2, t_1+t_2+\frac{1}{2}( x_1 y_2 - y_1 x_2) \right)$ . This group law commutes with the dilation $(x,y,t) \mapsto (r x, ry, r^2 t)$ , so $\mathbb{H}_i$ cannot have a Riemannian metric on it. Instead, it is equipped with a sub-Riemannian metric: an inner product on a distribution over $\mathbb{H}_i$ . At the origin, this distribution gives the XY plane, and the inner product is just the Euclidean inner product on this two-dimensional subspace. This horizontal distribution and inner product is defined at the other points using invariance under left multiplication. Here is the standard picture of the horizontal distribution. We just show what happens along the XY plane. Vertical translation by $(0,0,t)$ acts as the identity on the distribution. $Distribution on Heisenberg Space, generic view$
Last Updated on Friday, 08 May 2009 20:10
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