Geometry of rank 2 distributions with nonzero Wilczynski invariants*
Dedicated to Andrei Agrachev on the occasion of his 60th birthday
- 10.1080/14029251.2014.900985How to use a DOI?
- Abnormal extremals of distributions; self-dual curves in projective space; Wilczynski invariants; canonical frames; sl2 –representations
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.
- © 2014 The Authors. Published by Atlantis Press and Taylor & Francis
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- This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).
Cite this article
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 -