  A 3D almost-Riemannian manifold is a generalized Riemannian manifold defined
locally by 3 vector fields that play the role of an orthonormal frame, but
could become collinear on some set $\Zz$ called the singular set. Under the
Hormander condition, a 3D almost-Riemannian structure still has a metric space
structure, whose topology is compatible with the original topology of the
manifold. Almost-Riemannian manifolds were deeply studied in dimension 2. In
this paper we start the study of the 3D case which appear to be reacher with
respect to the 2D case, due to the presence of abnormal extremals which define
a field of directions on the singular set. We study the type of singularities
of the metric that could appear generically, we construct local normal forms
and we study abnormal extremals. We then study the nilpotent approximation and
the structure of the corresponding small spheres. We finally give some
preliminary results about heat diffusion on such manifolds.
