Almost Riemannian geometry from a control theory point of view.

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 two-dimensional 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 almost-Riemannian structures. They are special cases of
rank-varying sub-Riemannian structures,
which are naturally defined in terms of submodules of the space of
smooth vector fields on $M$.
Almost-Riemannian structures show
interesting phenomena, in particular for what concerns the
relation between curvature, presence of conjugate points, and topology of
the manifold.
The main result is a generalization to almost-Riemannian
structures of the Gauss-Bonnet formula.