The intrinsic geometry of the osculating structures that underlie the Heisenberg calculus (or Why the tangent space in sub-Riemannian geometry is a group)

Details The intrinsic geometry of the osculating structures that underlie the Heisenberg calculus (or Why the tangent space in sub-Riemannian geometry is a group) The intrinsic geometry of the osculating structures that underlie the Heisenberg calculus (or Why the tangent space in sub-Riemannian geometry is a group)

2010-07-27

arxiv.org/abs/1007.4759v1

Mathematics

Analysis of PDEs

23 pages

Scientific paper

We explore the geometry that underlies the osculating structures of the Heisenberg calculus. For a smooth manifold M with a distribution H in TM analysts have developed explicit (and rather complicated) coordinate formulas to define the nilpotent groups that are central to the calculus. Our aim is, specifically, to gain insight in the intrinsic structures that underlie these coordinate formulas. There are two key ideas. First, we construct a certain generalization of the notion of tangent vectors, called "parabolic arrows", involving a mix of first and second order derivatives. Parabolic arrows are the natural elements for the nilpotent groups of the osculating structure. Secondly, we formulate the natural notion of exponential map for the fiber bundle of parabolic arrows, and show that it explains the coordinate formulas of osculating structures. The result is a conceptual simplification and unification of the treatment of the Heisenberg calculus found in the analytic literature. As a bonus we obtain insight in how to construct a tangent groupoid for this calculus (for arbitrary filtered manifolds), which is a key tool in the study of hypoelliptic Fredholm index problems.

Affiliated with

Also associated with

No associations

Rating

The intrinsic geometry of the osculating structures that underlie the Heisenberg calculus (or Why the tangent space in sub-Riemannian geometry is a group) does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with The intrinsic geometry of the osculating structures that underlie the Heisenberg calculus (or Why the tangent space in sub-Riemannian geometry is a group), we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The intrinsic geometry of the osculating structures that underlie the Heisenberg calculus (or Why the tangent space in sub-Riemannian geometry is a group) will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-321564