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We explore the geometry that underlies the osculating nilpotent groupstructures of the Heisenberg calculus. For a smooth manifold $M$ with adistribution $H\subseteq TM$ analysts use explicit (and rather complicated)coordinate formulas to define the nilpotent groups that are central to thecalculus. Our aim in this paper is to provide insight in the intrinsic geometrythat underlies these coordinate formulas. First, we introduce `parabolicarrows' as a generalization of tangent vectors. The definition of parabolicarrows involves a mix of first and second order derivatives. Parabolic arrowscan be composed, and the group of parabolic arrows can be identified with thenilpotent groups of the (generalized) Heisenberg calculus. Secondly, weformulate a notion of exponential map for the fiber bundle of parabolic arrows,and show how it explains the coordinate formulas of osculating structures foundin the literature on the Heisenberg calculus. The result is a conceptualsimplification and unification of the treatment of the Heisenberg calculus.

The Geometry of the Osculating Nilpotent Group Structures of the Heisenberg Calculus