Heisenberg Group - As A Sub-Riemannian Manifold

As A Sub-Riemannian Manifold

The three-dimensional Heisenberg group H₃(R) on the reals can also be understood to be a smooth manifold, and specifically, a simple example of a sub-Riemannian manifold. Given a point p=(x,y,z) in R3, define a differential 1-form Θ at this point as

This one-form belongs to the cotangent bundle of R3; that is,

is a map on the tangent bundle. Let

It can be seen that H is a subbundle of the tangent bundle TR3. A cometric on H is given by projecting vectors to the two-dimensional space spanned by vectors in the x and y direction. That is, given vectors and in TR3, the inner product is given by

The resulting structure turns H into the manifold of the Heisenberg group. An orthonormal frame on the manifold is given by the Lie vector fields

which obey the relations =Z and ==0. Being Lie vector fields, these form a left-invariant basis for the group action. The geodesics on the manifold are spirals, projecting down to circles in two dimensions. That is, if

is a geodesic curve, then the curve is an arc of a circle, and

with the integral limited to the two-dimensional plane. That is, the height of the curve is proportional to the area of the circle subtended by the circular arc, which follows by Stokes' theorem.