Heisenberg Space

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.

$Distribution on Heisenberg Space, generic view$

Note that Heisenberg translations shear rather than rotate. So from above this picture looks like a checkerboard:

$Distribution on Heisenberg Space - top view$

The view from the side ((1,0,0) direction, no perspective) is a bit different though:

$Distribution on Heisenberg Space viewed from the side$

A curve in $\mathbb{H}_i$ is horizontal if its derivative at each point lies in the horizontal distribution. Integrating the speed of a horizontal curve (with respect to the sub-Riemannian inner product) gives its length, and the Carnot-Caratheodory (CC) distance between two points is defined as the infemum over the lengths of curves joining them. As it turns out, $(\mathbb{H}_i,CC)$ is geodesic: there is always a shortest path between any two given points. However, if the two points have the same xy coordinates, this path is not unique. Here is a picture of all the curves going up from the origin to $(0,0,1)$ :

$Bubble Set in Heisenberg Space$

Each of these geodesics projects to a circle in the XY plane, and the radius of the circle determines how high the geodesic climbs. These bubble sets are perhaps the most interesting surfaces in Heisenberg space, as they are the only surfaces of constant curvature and are conjectured to be the solution to the parametric inequality, which I'll describe later.

The spheres in the CC metric are strange objects that look like apples:

$CC sphere in Heisenberg space$

The spheres are neither totally geodesic surfaces, nor have constant curvature. They're also clearly not convex. However, geodesic convexity is not a very useful feature in Heisenberg space: since the bubble sets are not convex, neither is any proper subset of Heisenberg space with positive volume. Here's a bubble set connecting two points contained in another bubble set:

$CC Bubbles are not convex$

Here's another fun picture: a CC sphere with corresponding bubble sets inside it, going to the north and south poles:

$CC Sphere with corresponding bubble sets$

Now, apart from the fun pictures, what's so interesting (to Anton) about Heisenberg space?

A few things:
1) It's a nilpotent step-1 Lie group. And Lie groups have lots of fun properties that can be studied on the level of the Lie algebra. The geometry of a Lie group (especially everything geodesic) can be read off from the Lie properties of its algebra, since the metric on the group is given by the Killing form on the Lie algebra.
2) It's the boundary of complex hyperbolic space. Well, almost. There's the point at infinity missing, but if you add that in, you get a contact structure on the 3-sphere that's invariant under hyperbolic isometries. And if you want to study a hyperbolic space, you definitely want to look at its boundary.
3) There might be real-world applications! In the Heisenberg group, you have two directions you can move in, and 3 dimensions that you can get to. That sort of thing happens all the time in applications. For example, if you're trying to parallel-park a car, you can move in two directions - rotate the wheel and go forward/back but want to move in three dimesions: X,Y, and the rotation of the car. There should be a way to describe the parallel parking problem in terms of the Heisenberg group. So this could help my parking skills.

I'll try to describe all three of these in more detail later on.

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Last Updated ( Friday, 08 May 2009 14:10 )