[Paper Review] Geometry of rank 2 distributions with nonzero Wilczynski invariants and affine control systems with one input

Boris Doubrov, Igor Zelenko|arXiv (Cornell University)|Jan 13, 2013

Advanced Algebra and Geometry26 references3 citations

TL;DR

This paper develops a canonical frame construction for rank 2 distributions of maximal class in $\mathbb{R}^n$ with non-vanishing generalized Wilczynski invariants and for affine control systems with one input, leveraging the canonical parametrization of abnormal extremals. By exploiting $\mathfrak{sl}_2$-representations and projective invariants of curves, it establishes a $ (2n-3) $-dimensional canonical frame independent of nilpotent approximation, with moduli spaces of symmetric models fully characterized for $ n \geq 5 $. The key contribution is a unified, invariant geometric framework for such systems using symplectic geometry of Jacobi curves.

ABSTRACT

We demonstrate how the novel approach to the local geometry of structures of nonholonomic nature, originated by Andrei Agrachev, works in the following two situations: rank 2 distributions of maximal class in R^n with non-zero generalized Wilczynski invariants and rank 2 distributions of maximal class in R^n with additional structures such as affine control system with one input spanning these distributions, sub-(pseudo)Riemannian structures etc. The common feature of these two situations is that each abnormal extremal (of the underlying rank 2 distribution) possesses a distinguished parametrization. This fact allows one to construct the canonical frame on a (2n-3)-dimensional bundle in both situations for arbitrary n greater than 4. The moduli spaces of the most symmetric models for both situations are described as well. The relation of our results to the divergence equivalence of Lagrangians of higher order is given

Motivation & Objective

• To develop a canonical frame for rank 2 distributions of maximal class in $\mathbb{R}^n$ with non-vanishing generalized Wilczynski invariants, extending beyond the zero-invariant case.
• To establish a geometric framework for affine control systems with one input by leveraging the canonical parametrization of abnormal extremals.
• To characterize the moduli space of the most symmetric models in both settings using symplectic invariants and $\mathfrak{sl}_2$-representations.
• To unify the construction of canonical frames across different geometric structures—distributions, sub-Riemannian, and control systems—by exploiting the Jacobi curve's differential geometry.

Proposed method

• The method uses the Jacobi curve of an abnormal extremal, which is a curve in the Lagrangian Grassmannian, to extract symplectic invariants.
• It applies the theory of projective structures on curves in $\mathbb{R}P^{n-2}$, particularly the existence of a canonical parametrization up to Möbius transformations.
• The construction relies on the fact that non-vanishing generalized Wilczynski invariants allow for a distinguished parametrization of abnormal extremals, replacing the projective structure.
• The canonical frame is built on a $ (2n-3) $-dimensional bundle using $\mathfrak{sl}_2$-representation theory and the structure of the Jacobi curve's symplectic curvature.
• Key equations include the bracket relations such as $[e, \varepsilon_1] = -\frac{1}{2}\varepsilon_1$ and $[h, \varepsilon_{2m}] = \sum_{i=1}^{m-1} (-1)^{i+1} r_i \varepsilon_{2(m-i)}$, which determine the frame's structure.
• The method is independent of the nilpotent approximation and growth vector, relying instead on the intrinsic geometry of the Jacobi curve and its symplectic invariants.

Experimental results

Research questions

• RQ1 How can a canonical frame be constructed for rank 2 distributions of maximal class in $\mathbb{R}^n$ when generalized Wilczynski invariants are non-zero, and what is the minimal dimension of the frame bundle?
• RQ2 What is the role of the canonical parametrization of abnormal extremals in replacing the projective structure for non-vanishing invariants?
• RQ3 How does the canonical frame construction for affine control systems with one input differ from the standard Tanaka prolongation when generalized Wilczynski invariants are non-zero?
• RQ4 What is the moduli space of the most symmetric models in both the distribution and control system settings, and how is it parametrized?
• RQ5 To what extent is the canonical frame construction independent of the nilpotent approximation and growth vector of the distribution?

Key findings

• A canonical frame is constructed on a $ (2n-3) $-dimensional bundle for rank 2 distributions of maximal class in $\mathbb{R}^n$ with non-vanishing generalized Wilczynski invariants, for all $ n \geq 5 $.
• The construction relies on the existence of a canonical parametrization of abnormal extremals, which replaces the projective structure when invariants are non-zero.
• The moduli space of the most symmetric models is described as a space of tuples $ (r_1, \ldots, r_{n-3}) $, where the $ r_i $ are the symplectic curvatures derived from the Jacobi curve.
• The canonical frame's structure functions are fully determined by the tuple $ (r_1, \ldots, r_{n-3}) $, with all nontrivial brackets expressible in terms of these constants.
• The frame is independent of the nilpotent approximation and growth vector, making it applicable to a broader class of distributions than the classical Tanaka method.
• For affine control systems with one input, the regularity condition ensures that the canonical parametrization of abnormal extremals is well-defined and leads to a unique canonical frame on the same $ (2n-3) $-dimensional bundle.