Subriemannian geometry studies a manifold endowed with a subbundle of its tangent bundle, also called a `distribution', and an inner product (metric) on that distribution. Thus you are constrained to move about in certain directions --only those tangent to the distributions.
This was my main area of work just before I got sucked in (gladly) to the 3-body problem. I've written a book on it
A Tour of Subriemannian Geometries, Their Geodesics and Applications available via the AMS.

More:
Pictures of various 3D subriemannian balls. by Monique Chyba

My review of Gromov's book

Abnormal Minimizers The first pf of existence of singular minimizers in sR geometry.

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

Engel and Goursat papers. These are remarkable families 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)