 
Summary: arXiv:math.OC/0609566v120Sep2006
A GaussBonnetlike Formula on TwoDimensional
AlmostRiemannian Manifolds1
Andrei A. Agrachev, Ugo Boscain
SISSAISAS, Via Beirut 24, 34014 Trieste, Italy
agrachev(at)sissa.it, boscain(at)sissa.it
Mario Sigalotti
Institut ´Elie Cartan, UMR 7502 NancyUniversit´e/CNRS/INRIA, POB 239, 54506 Vandoeuvrel`esNancy, France
Mario.Sigalotti(at)inria.fr
21st September 2006
Abstract  We consider a generalization of Riemannian geometry that naturally
arises in the framework of control theory. Let X and Y be two smooth vector fields
on a twodimensional manifold M. If X and Y are everywhere linearly independent,
then they define a classical Riemannian metric on M (the metric for which they are
orthonormal) and they give to M the structure of metric space. If X and Y become
linearly dependent somewhere on M, then the corresponding Riemannian metric has
singularities, but under generic conditions the metric structure is still well defined.
Metric structures that can be defined locally in this way are called almostRiemannian
structures. They are special cases of rankvarying subRiemannian structures, which
are naturally defined in terms of submodules of the space of smooth vector fields on
