I will start with basic definitions.

So M will be 3D manifold and equipped with a contact structure, which is a sub-bandal,

a round-to-sub-bandal of the tangent bandal.

A smooth family of subspecies of the tangent bandal, which is defined as a kernel of one

form.

And the main property is that alpha wedge d alpha is a volume form, though alpha is

non-degenerate in some sense, and though M is orientable in particular.

Then we can define the Reb vector field associated to alpha.

The contact structure, alpha is not part of the contact structure.

You can change alpha, multiply alpha by some function.

But if you give alpha, there is a well-defined vector field called the Reb vector field,

which is defined by, if you evaluate alpha on this vector field, you get 1.

And you take the inner product of the Reb vector field with d alpha, you get 0.

So this is given a uniquely defined vector field, which is transversal to the distribution.

Here is George Reb.

And this vector field has an Hamiltonian interpretation.

What you do, you look at the symplectic, at the orthogonal of the distribution, which

is a sub-con of the cotangent space.

So you look at one form which vanish on d.

And you define the Hamiltonian rho on sigma, which is simply rho of s times alpha is absolute

value of s.

So this is given Hamiltonian dynamics on sigma, because sigma is a symplectic manifold.

And this dynamics is homogeneous of the 0.

And it projects onto the Reb orbit, to its kind of lift of the Reb dynamics to a sub-con

of the cotangent space.

And we denote by RT the Reb flow on sigma.

Or because it's invariant by its homogeneous of the 0, you can look at this field on s

of sigma, which is a sphere bounded.

S of sigma is simply the quotient of sigma by homothesis.

Now, till now there is no metric, but I will give a metric.

So I give a smooth metric J on d, which is like a Riemannian metric, but not on the full

tangent space, but only on the subspace.

This defines what is called a Subriemannian manifold, and we get a distance on the manifold

by minimizing the length of what is called horizontal curves, which are curves tangent

to the distribution at every point.

And for this metric, there are geodesics, which are locally minimizing the distance.

And as usual, the geodesics can be defined by an Hamiltonian, which is simply given here.

You look at cotangent vector xi.

You restrict xi to the distribution at the point x, and you evaluate the norm with respect

to J. Give you what is called the co-metric in the Subriemannian geometric.

And you look at the dynamics of J star inside the unit cotangent bundle as in Riemannian

geometric.

So the same Riemannian geometry, you get the geodesics.

Here you get an Hamiltonian system on the cotangent space on the unit bundle, which

was, if you project the orbits onto M, it's the M.

You get the geodesics.

And we can assume that the distribution day is oriented.

And from this, we get a contact form, which is defined by asking that the differential

of alpha J, which is a two form when you restrict today is given by the volume form of J.

From this, we get a rib vector field associated to this one form.