Piotr Mormul

Goursat Flags: Classification of Codimension-One Singularities

AbstractA distribution D of corank

On local classification of Goursat structures

r \geqslant 2

Modules of vector fields, differential forms and degenerations of differential systems

on a manifold M is Goursat when its Lie square [D, D] is a distribution of constant corank r-1, the Lie square of [D, D] is of constant corank r-2 and so on. Any such D, according to von Weber [21] and E. Cartan [3], behaves in a well-known way at generic points of M: in certain local coordinates it is the chained model (C) given below, a classical object in the control theory. Singularities concealed in Goursat distributions have emerged for the first time in [8]; by now the complete local classification of these objects of coranks not exceeding 7 is known, plus some isolated facts for coranks 8, 9, and 10. In the present paper we deal with the Goursat distributions of any corank r and obtain a complete classification of the first occurring singularities of them, located at points outside a stratified codimension-2 submanifold of M. Off this set there are (on top of (C)) only r-2 non-equivalent singular behaviours possible.

Singularity Classes of Special 2-Flags

Goursat distributions are subbundles, of codimension at least 2, in the tangent bundles to manifolds having the flag of consecutive Lie squares of ranks not depending on a point and growing—very slowly—always by 1. The length of a flag thus equals the corank of the underlying distribution.After the works of, among others, Bryant&Hsu (1993), Jean (1996), and Montgomery&Zhitomirskii (2001), the local behaviours of Goursat flags of any fixed length r≥2 are stratified into geometric classes encoded by words of length r over the alphabet {G,S,T} (Generic, Singular, Tangent) starting with two letters G and having letter(s) T only directly after an S, or directly after another T.It follows from [6] that the Goursat germs sitting in any fixed geometric class have, up to translations by rk D−2, one and the same small growth vector (at the reference point) that can be computed recursively in terms of the G,S,T code. In the present paper we give explicit solutions to the recursive equations of Jean and show how, thanks to a surprisingly neat underlying arithmetics, one can algorithmically read back the relevant geometric class from a given small growth vector. This gives a secondary, Gödel-like super-encoding of the geometric classes of Goursat objects (rather than just a 1-1 correspondence between those classes and small growth vectors).

Do Moduli of Goursat Distributions Appear on the Level of Nilpotent Approximations

Abstract Goursat structures are the dual objects of E. Cartans “ systems of class zero ” (also called “ en drapeau ” in [7]). We prove that the growth vector is not the only local invariant for a Goursat structure on ℝ n , n ≥ 9; and that there exist infinitely many locally non-equivalent Goursat structures on ℝ n , n ≥ 10.

RANK 2 DISTRIBUTIONS SATISFYING THE GOURSAT CONDITION: ALL THEIR LOCAL MODELS IN DIMENSION 7 AND 8

We first give a proof of a result announced in [16] that Goursat distributions of arbitrary corank (= length of the associated flag of consecutive Lie squares of a G. distribution) locally possess nilpotent bases (i. e., bases generating over R nilpotent Lie algebras) of explicitly computable orders of nilpotency of the induced Lie algebras (KR algebras). We say that G. distributions are locally weakly nilpotent in the sense of [11]. Recalling that the germs of such distributions are stratified into geometric classes of Jean, Montgomery and Zhitomirskii, in certain geometric classes termed tangential, the computed nilpotency orders of KR algebras turn out to coincide with the nonholonomy degrees, computed by Jean, at the reference points for germs. In the tangential classes, then, the nilpotency orders of KR algebras are minimal among all possible nilpotent bases. Secondly, in dimension 6 and 7, two smallest dimensions in which not all geometric classes are tangential, we show that all G. germs in the non-tangential classes are not strongly nilpotent in the sense of [2].

Multi-dimensional Cartan prolongation and special k–flags

AbstractWe deal with (n−1)-generated modules of smooth (analytic, holomorphic) vector fieldsV=(X1,..., Xn−1) (codimension 1 differential systems) defined locally on ℝn or ℂn, and extend the standard duality(X1,..., Xn−1)↦(ω), ω=Ω(X1,...,Xn−1,.,) (Ω−a volume form) betweenV′s and 1-generated modules of differential 1-forms (Pfaffian equations)—when the generatorsXi are linearly independent—onto substantially wider classes of codimension 1 differential systems. We prove that two codimension 1 differential systemsV and