Geometric classes of Goursat flags and the arithmetics of their encoding by small growth vectors

Piotr Mormul. "Geometric classes of Goursat flags and the arithmetics of their encoding by small growth vectors." Open Mathematics 2.5 (2004): 859-883. <http://eudml.org/doc/268896>.

@article{PiotrMormul2004,
abstract = {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).},
author = {Piotr Mormul},
journal = {Open Mathematics},
keywords = {58A17},
language = {eng},
number = {5},
pages = {859-883},
title = {Geometric classes of Goursat flags and the arithmetics of their encoding by small growth vectors},
url = {http://eudml.org/doc/268896},
volume = {2},
year = {2004},
}

TY - JOUR
AU - Piotr Mormul
TI - Geometric classes of Goursat flags and the arithmetics of their encoding by small growth vectors
JO - Open Mathematics
PY - 2004
VL - 2
IS - 5
SP - 859
EP - 883
AB - 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).
LA - eng
KW - 58A17
UR - http://eudml.org/doc/268896
ER -