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
a great list of basic sR and metric geometry references
compiled by Enrico LeDonne.
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. )
-- 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.