Minimal nilpotent algebras in Goursat flags of lengths not exceeding 6
by
Piotr Mormul
Institute of Mathematics, Warsaw University, Warsaw, Poland

Goursat distributions are special rank-2 subbundles, D, in the tangent bundle to an n-dimensional manifold (n >= 4) such that the sequence of consecutive Lie squares of D consists of regular distributions of ranks 3, 4, ...,  n-1,  n. The FLAG of D is the sequence of those Lie squares, including D. Its length (or: the corank of D) is r = n-2.

In the abstract [M2] we recalled a basic partition of Goursat germs into disjoint GEOMETRIC CLASSES encoded by words of length r >= 2 over the alphabet {G, S, T}, with two first letters always G and such that never a T goes directly after a G. Their construction was done explicitly by Montgomery and Zhitomirskii; it is reproduced in Section 1.3 of [M1]. (Implicitly these classes are already present in the pioneering work [J]. Another way of constructing them has been proposed in [PR1] where they are called singularity types.)

In dimension 4 (length 2) there is but one geometric class GG, in dimension 5 (length 3) - only GGG and GGS, in dimension 6 (length 4) - GGGG, GGSG, GGST, GGSS, GGGS. Generally, for length r, there are u_2r-3 (Fibonacci number) geometric classes.

The union of all geometric classes of fixed length r, with letters S in fixed positions in the codes, is called, after [MonZh], a Kumpera-Ruiz class of Goursat germs of that corank r (remembering the work [KR] dealing systematically with singularities of Goursat distributions). For instance, in length 4, the two geometric classes GGSG and GGST build one KR class **S*. Naturally, the number of KR classes of length r is 2^r-2, and they are pairwise disjoint, too.

We call TANGENTIAL geometric classes whose codes possess letters G only in the beginning, before the first S (if any) in the code. Because they sit in their respective KR classes (GGST sits in **S*, GGGS in ***S, etc), there is a bijection between the tangential and KR classes. So there are 2^r-2 tangential classes of length r. Up to length 3 all classes are tangential (2⁰ = u₁, 2¹ = u₃). The unique non-tangential class in length 4 is GGSG. For general length r there are u_2r-3 - 2^r-2 non-tangential geometric classes: five for length 5 (GGSGG, GGSTG, GGSSG, GGSGS, GGGSG), eighteen for length 6, fifty-seven for length 7, ...

Nilpotent local bases, existing only for certain distributions, were addressed systematically in [HLS] and in several posterior works. In [M3] computed were the nilpotency orders for the [now known to be nilpotent] bases of G. germs put forward in [KR]. (We mean the nilpotency orders of the Lie algebras generated by those bases.) Such nilpotency order, written symbolically NO, is one and the same within any fixed KR class. We stress that, however constant over a huge KR class, every NO is attached to a concrete local basis of a concrete G. germ and a priori does NOT have an intrinsic character. (In [PR2] discussed was the existence of nilpotency orders - not explicit formulas for them - for those same local bases of G.)

On the other hand, in [J] Jean was able to compute the nonholonomy degrees (the lengths, written NH, of small growth vectors for completely nonholonomic distributions) for germs showing up in his trigonometric presentation of Goursat objects. That - in view of Theorem 4.1 of [BH] and after putting the geometric classes in relief in Jean's approach - gives the values of NH for all G. germs. The nonholonomy degree appears to be one and the same within each fixed geometric class.

Whatever local nilpotent basis with its pertinent value of NO, for whatever completely nonholonomic distribution, there always holds the inequality NH <= NO. And the formulas of [J] and [M3], when compared, comply, for G. objects, with this general rule. By inspecting those formulas closer, NH = NO only when the relevant geometric class is tangential. Thus within the tangential classes the KR bases cannot be improved in the sense of lowering nilpotency orders (so within tangential, the NO's proposed in [M3] do have invariant geometric character).

What does it happen in non-tangential classes (when NH < NO) ? Do there exist G. germs with better nilpotent bases - with nilpotency orders smaller than NO's in the relevant KR classes?

This question makes sense for lengths allowing for non-tangential classes, i.e., from 4 (dimension 6) onwards. We announce below the full answer for lengths 4 and 5 (addressing all non-tangential classes in these lengths), and a partial answer in length 6 (addressing eight out of u₉ - 2⁴ = 18 non-tangential classes).

Theorem.

A. In dimension 6, for Goursat germs in the class GGSG the nilpotency order NO = 7 is minimal among all possible local nilpotent bases, despite the fact that NH = 6 for these germs.

B. In dimension 7, for germs in the classes (in parenthes given are the values of NH and NO)

GGSGG   (7,    9)

GGSTG   (8,    9)

GGSSG   (9,    11)

GGSGS   (11,    12)

GGGSG   (8,    10)

their respective KR bases generate nilpotent Lie algebras of minimal possible nilpotency orders - these listed values of NO.

C. In dimension 8, for germs in the non-tangential classes (NH and NO are given in parentheses next to the class)

GGSGGG   (8,    11)

GGSTGG   (9,    11)

GGSTTG   (10,    11)

GGSSGS   (17,    19)

GGSGSG   (12,    17)

GGSTSG   (13,    17)

GGSGST   (16,    17)

GGGSGS   (15,    17)

their respective KR bases are of minimal possible nilpotency order, too.

(We do not yet know the answer in the remaining ten non-tangential classes in length 6.) Note that even when NO exceeds considerably NH (in part C. of Theorem), the KR bases nonetheless turn out optimal for the relevant G. germs. It appears a posteriori that in the cases discussed in Theorem the orders NO are, after all, intrinsically attached to the respective non-tangential germs.

R E F E R E N C E S

[BH] R. Bryant, L. Hsu; Rigidity of integral curves of rank 2 distributions. Invent. math. 114 (1993), 435 - 461.

[HLS] H. Hermes, A. Lundell, D. Sullivan; Nilpotent bases for distributions and control systems. J. Diff. Eqns 55 (1984), 385 - 400.

[J] F. Jean; The car with N trailers: characterisation of the singular configurations. ESAIM: Control, Optimisation and Calculus of Variations (URL: http://www.emath.fr/cocv/) 1 (1996), 241 - 266.

[KR] A. Kumpera, C. Ruiz; Sur l'equivalence locale des systemes de Pfaff en drapeau. In: F. Gherardelli (ed.), Monge-Ampere Equations and Related Topics

Date received: April 21, 2002

Copyright © 2002 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cajc-15.