Subriemannian geometry , also called Carnot-Caratheodory geometry is the differential and metric geometry induced on an n-manifold by insisting that one's motion at each point is restricted to lie in a k-plane. More precisely, one is given a ``bracket generating'' (*) distribution D , which is a linear sub-bundle of Q's tangent bundle, and a fiber inner product on D. A smooth path is called `horizontal' if it is tangent to D. The length of such a path can be computed using the inner product. The distance between two points of Q is the infimum of the lengths of the horizontal points connecting the two points. The first non-trivial example of a subRiemannian manifold is the 3 dimensional Heisenberg group endowed with the generators X, Y of the Heisenberg algebra as an orthonormal basis for the horizontal distribution. A general feature of subRiemannian geometry is that the manifold dimension is always less than the (metric) Hausdorff dimension. The Heisenberg group has Hausdorff dimension 4. I worked for about 5 years in SubRiemannian geometry and moved mostly away from it when I got sucked in working on the 3-body problem.

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.