We explore the geometry that underlies the osculating nilpotent group
structures of the Heisenberg calculus. For a smooth manifold M with a
distribution H⊆TM analysts use explicit (and rather complicated)
coordinate formulas to define the nilpotent groups that are central to the
calculus. Our aim in this paper is to provide insight in the intrinsic geometry
that underlies these coordinate formulas. First, we introduce `parabolic
arrows' as a generalization of tangent vectors. The definition of parabolic
arrows involves a mix of first and second order derivatives. Parabolic arrows
can be composed, and the group of parabolic arrows can be identified with the
nilpotent groups of the (generalized) Heisenberg calculus. Secondly, we
formulate a notion of exponential map for the fiber bundle of parabolic arrows,
and show how it explains the coordinate formulas of osculating structures found
in the literature on the Heisenberg calculus. The result is a conceptual
simplification and unification of the treatment of the Heisenberg calculus.Comment: Some parts rewritten; section 3 added. 33 page