I wrote a book titled A Tour of Subriemannian Geometries, Their Geodesics and Applications on the subject

Here is a great list of basic sR and metric geometry references compiled by Enrico LeDonne.

Gromov has a book on sR geometry, and also wrote this, for context .

One of my favorite references, which also wonderfully explains the Cartan method of equivalenc is: Keener Hughen's thesis in which he describes the complete invariants for subRiemannian geometries of contact type on 3-manifolds (**).

Here a few of my papers on the subject. (These can all be found here also. )

Abnormal Minimizers -- the first proof of existence of a singular minimizer in sR geometry.

Nonintegrable sR geodesic flow on a Carnot group with M. Shapiro, A. Stolin in JDCS 3, 4 1997, 519-530

The next few papers concern Engel and Goursat distributions which are remarkable kinds of rank 2 distributions. The Engel distributions are `stable': they admit a Darboux theorem, and are the only stable distributions outside of the contact and quasi-contact distributions.

``GEOMETRIC APPROACH TO GOURSAT FLAGS'', with Michail Zhitomirskii, (postscript)

``Engel deformations and contact structures'' ( postscript)

My review of Gromov's book

(*) The `bracket generating' condition is also called ``Hormander's condition. The Chow-Rashevskii theorem asserts that if the distribution is bracket generating and the manifold is connected then any two points can be joined by a horizontal path.

(**) K. Hughen finds two invariants. One is the `Gauss curvature' along the 2-distribution. The other is a `torsion' which measures the failure of the Reeb vector field to generate a 1-parameter group of subRiemannian isometry. Hughen also proves a Bonnet-Myers type theorem for geometries with appropriate bounds placed on these two curvatures. This thesis is also a great way to get into the Cartan machinery.