  This paper studies the infinitesimal structure of Carnot manifolds. By a
Carnot manifold we mean a manifold together with a subbundle filtration of its
tangent bundle which is compatible with the Lie bracket of vector fields. We
introduce a notion of differential, called Carnot differential, for Carnot
manifolds maps (i.e., maps that are compatible with the Carnot manifold
structure). This differential is obtained as a group map between the
corresponding tangent groups. We prove that, at every point, a Carnot manifold
map is osculated in a very precise way by its Carnot differential at the point.
We also show that, in the case of maps between nilpotent graded groups, the
Carnot differential is given by the Pansu derivative. Therefore, the Carnot
differential is the natural generalization of the Pansu derivative to maps
between general Carnot manifolds. Another main result is a construction of an
analogue for Carnot manifolds of Connes' tangent groupoid. Given any Carnot
manifold $(M,H)$ we get a smooth groupoid that encodes the smooth deformation
of the pair $M\times M$ to the tangent group bundle $GM$. This shows that, at
every point, the tangent group is the tangent space in a true
differential-geometric fashion. Moreover, the very fact that we have a groupoid
accounts for the group structure of the tangent group. Incidentally, this
answers a well-known question of Bella\"iche.
