Geometry of rank 2 distributions with nonzero Wilczynski invariants*

DOI

10.1080/14029251.2014.900985How to use a DOI?

Keywords

Abnormal extremals of distributions; self-dual curves in projective space; Wilczynski invariants; canonical frames; sl2 –representations

Abstract

In the famous 1910 “cinq variables” paper Cartan showed in particular that for maximally nonholonomic rank 2 distributions in ℝ⁵ with non-zero covariant binary biquadratic form the dimension of the pseudo-group of local symmetries does not exceed 7 and among such distributions he described the one-parametric family of distributions for which this pseudo-group is exactly 7-dimensional. Using the novel interpretation of the Cartan covariant binary biquadratic form via the classical Wilczynski invariant of curves in projective spaces associated with abnormal extremals of the distributions [4, 27, 28] one can generalize this Cartan result to rank 2 distributions in ℝⁿ satisfying certain genericity assumption, called maximality of class, for arbitrary n ≥ 5.

In the present paper for any rank 2 distribution of maximal class with at least one nonvanishing generalized Wilczynski invariants we construct the canonical frame on a (2n — 3)-dimensional bundle and describe explicitly the moduli spaces of the most symmetric models. The relation of our results to the divergence equivalence of Lagrangians of higher order is given as well.

Copyright

© 2014 The Authors. Published by Atlantis Press and Taylor & Francis

Open Access

This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - Boris Doubrov
AU  - Igor Zelenko
PY  - 2021
DA  - 2021/01/06
TI  - Geometry of rank 2 distributions with nonzero Wilczynski invariants*
JO  - Journal of Nonlinear Mathematical Physics
SP  - 166
EP  - 187
VL  - 21
IS  - 2
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2014.900985
DO  - 10.1080/14029251.2014.900985
ID  - Doubrov2021
ER  -