Journal of Lie Theory

We explore the geometry that underlies the osculating nilpotent group structures of the Heisenberg calculus. For a smooth manifold M with a distribution H contained in 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 clarifies the coordinate formulas of osculating structures found in the literature on the Heisenberg calculus.