Geometric singularity classes for special k-flags, k >= 2, of arbitrary length
by
Piotr Mormul
Institute of Mathematics, Warsaw University, Warsaw, Poland

1. DEFINITION OF SPECIAL k-FLAGS AND THE AIM:

Special k-flags (k >= 2) of all lengths r >= 1 have been defined in [M] by conditions formally stronger than the conditions defining in [PR] 'generalized contact systems for curves', or else than those of [KRub] putting in evidence 'k-flags satisfying certain normality conditions'. The reason was that precisely such conditions were prompted by one structural theorem in [BH] (attributed by Bryant and Hsu to E.Cartan) that, in [M], was generalized by means of multi-dimensional Cartan prolongations.

Under closer inspection, using two early Bryant's results quoted in [PR] as well as one original lemma of the authors of [PR], the two definitions (or three, taking into account also that of [KRub]) boil down to the same.

Namely, the tower of consecutive Lie squares of D

TM = D0 contains D1 contains D2 contains ... contains Dr-1 contains Dr = D

(that is, D^j-1 = [D^j, D^j] for j = r, r-1, ..., 2, 1) should consist of distributions of ranks, starting from the smallest object D^r: k+1, 2k+1, ..., rk + 1, (r+1)k + 1 = dimM such that

* for j = 1, ..., r-1 the Cauchy-characteristic module L(D^j) of D^j sits already in the smaller object D^j+1: L(D^j) subset D^j+1 and is regular of corank 1 in D^j+1, while L(D^r) = 0 ;

* * the covariant subdistribution F of D¹ (see [KRub], p.4 for the definition extending the classical Cartan approach of 1910, cf. also [MPel]) is involutive and of corank 1 in D¹ (hence also regular).this additional requirement 'corank 1 for F in D¹' is superfluous once that covariant object is assumed to be involutive, cf. Lem.1 in [KRub]ocal polynomial pseudo-normal forms, so-called EKR's (Extended Kumpera Ruiz) for such D were started in [KRub] and [PR], then fully constructed only in [M], after which a question had appeared about the geometrical meaning and significance of different families of those pseudo-normal forms.

Just like a similar question for Goursat flags [that is - outside the scope of the present abstract - when k = 1; the definition of 1-flags is more compact and simpler] pending several years after [KRui], started to be settled only in [BH], leading eventually, for any fixed length r, to 2^r-2 Kumpera-Ruiz singularity classes in [MonZ], encoded by the words over {*, S} and exactly corresponding to the original pseudo-normal forms of [KRui].

Footnotes:

this additional requirement 'corank 1 for F in D¹' is superfluous once that covariant object is assumed to be involutive, cf. Lem.1 in [KRub]L

Date received: April 26, 2003