Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Prolongations of $(3, 6)$-distributions by singular curves

Goo Ishikawa, Yoshinori Machida

TL;DR

This work studies how singular curves (abnormal extremals) drive a hierarchical prolongation of a $(3,6)$-distribution on a 6-manifold into new geometric structures: first a $(3,5,7,8)$-distribution with a $B_3(2,3)$ pseudo-product, then an iterated prolongation to a $(3,5,7,8,9)$-distribution with a $B_3(1,2,3)$ structure; it also analyzes an alternative $(4,6,8)$-prolongation with a generalized $B_3(1,3)$ structure. The main theorem establishes natural bijections between local isomorphism classes of these four classes, compatible with the prolongation/reduction operations, effectively unifying their classification and connecting to classical $B_3$-models realized on flag manifolds in $\,\mathbb{R}^{3,4}$. Throughout, the authors develop and exploit the singular velocity cones and the Tanaka-style pseudo-product structures to relate the different prolongation pathways. The results extend the understanding of how Hamiltonian/abnormal-geometry data control high-step prolongations in parabolic/Cartan-type geometries and illuminate links to Bryant-type conformal structures via the $B_3$-framework.

Abstract

A subbundle of rank 3 in the tangent bundle over a 6-dimensional manifold is called a (3, 6)-distribution if its local sections generate the whole tangent bundle by taking their Lie brackets once. An integral curve of a distribution, whose velocity vectors belong to the distribution, can be a singular curve or an abnormal extremal in the sense of geometric control theory. In this paper, given a (3, 6)-distribution, we prolong it, using the data of singular curves, to a (3,5,7,8)-distribution, to a (3, 5, 7, 8, 9)-distribution which possesses additional pseudo-product structure respectively. Regarding also another prolongation to a (4, 6, 8)-distribution, we show the equivalence of the classification problems of those four classes of distributions obtained from (3, 6)-distributions, generalising the correspondences of those in B_3-SO(3,4)-homogeneous models.

Prolongations of $(3, 6)$-distributions by singular curves

TL;DR

This work studies how singular curves (abnormal extremals) drive a hierarchical prolongation of a -distribution on a 6-manifold into new geometric structures: first a -distribution with a pseudo-product, then an iterated prolongation to a -distribution with a structure; it also analyzes an alternative -prolongation with a generalized structure. The main theorem establishes natural bijections between local isomorphism classes of these four classes, compatible with the prolongation/reduction operations, effectively unifying their classification and connecting to classical -models realized on flag manifolds in . Throughout, the authors develop and exploit the singular velocity cones and the Tanaka-style pseudo-product structures to relate the different prolongation pathways. The results extend the understanding of how Hamiltonian/abnormal-geometry data control high-step prolongations in parabolic/Cartan-type geometries and illuminate links to Bryant-type conformal structures via the -framework.

Abstract

A subbundle of rank 3 in the tangent bundle over a 6-dimensional manifold is called a (3, 6)-distribution if its local sections generate the whole tangent bundle by taking their Lie brackets once. An integral curve of a distribution, whose velocity vectors belong to the distribution, can be a singular curve or an abnormal extremal in the sense of geometric control theory. In this paper, given a (3, 6)-distribution, we prolong it, using the data of singular curves, to a (3,5,7,8)-distribution, to a (3, 5, 7, 8, 9)-distribution which possesses additional pseudo-product structure respectively. Regarding also another prolongation to a (4, 6, 8)-distribution, we show the equivalence of the classification problems of those four classes of distributions obtained from (3, 6)-distributions, generalising the correspondences of those in B_3-SO(3,4)-homogeneous models.
Paper Structure (9 sections, 15 theorems, 85 equations)

This paper contains 9 sections, 15 theorems, 85 equations.

Table of Contents

  1. Introduction.
  2. Singular curves of distributions.
  3. Singular curves of $(3, 6)$-distributions
  4. Prolongation of $(3, 6)$-distributions by singular curves.
  5. Iterated-prolongation of a $(3, 6)$-distribution.
  6. Another prolongation of a $(3, 6)$-distribution.
  7. Another iterated prolongation of a $(3, 6)$-distribution.
  8. Proof of the main theorem.
  9. Appendix: $B_3$-models.

Key Result

Theorem 1.1

There are bijective correspondences, compatible with prolongation procedures, between local isomorphism classes of $(3, 6)$-distributions, those of $(3, 5, 7, 8)$-distributions with pseudo-product structure of type $B_3(2,3)$, those of $(3, 5, 7, 8, 9)$-distributions with pseudo-product structure of

Theorems & Definitions (32)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 3.1
  • Definition 4.1
  • Lemma 4.2
  • Remark 4.3
  • Proposition 4.4
  • Lemma 4.5
  • Proposition 4.6
  • Example 4.7
  • ...and 22 more