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.