Subriemannian geometry also known as Carnot-Caratheodory geometry, concerns the geometry around a manifold endowed with a fiber inner product on a linear subbundle of its tangent bundle, Riemannian geometry being the `special case' where this subbundle is the entire tangent bundle. (0)

A Tour of Subriemannian Geometries, Their Geodesics and Applications , my book.
Gromov's book
context for Gromov.
Keener Hughen's thesis describing complete invariants for subRiemannian geometries of contact type on 3-manifolds. (1)

A few of my papers:
Abnormal Minimizers 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)


A few failed links I hope to reconnect to soon (written Dec 31, 2011)
Pictures of various 3D subriemannian balls. by Monique Chyba

My review of Gromov's book

(0) The chosen subbundle of the tangent bundle is called the `horizontal distribution' or just `distribution'. As you move about the manifold, your motion is constrained to be horizontal, meaning tangent to the horizontal distribution. One assumes the distribution is `bracket generating' which implies, by the Chow-Rashevskii theorem, that you can still move from any point to any other. Thus you are constrained to move about in certain directions --only those tangent to the distributions. 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 mostly in SubRiemannian geometry just before I got sucked in to the 3-body problem.
(1) 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.