Multi-dimensional Cartan prolongation and special k–flags

Abstract

Since the mid-nineties it has gradually become understood that the Cartan prolongation of rank 2 distributions is a key operation leading locally, when applied many times, to all so-called Goursat distributions. That is those, whose derived flag of consecutive Lie squares is a 1-flag (growing in ranks always by 1). We first observe that successive generalized Cartan prolongations (gCp) of rank k+1 distributions lead locally to all special k-flags: rank k+1 distributions D with the derived flag F being a k-flag possessing a corank 1 involutive subflag preserving the Lie square of F . (Note that 1-flags are always special.)Secondly, we show that special k-flags are effectively nilpotentizable (or: weakly nilpotent) in the sense that local polynomial pseudo-normal forms for such D resulting naturally from sequences of gCp's give local nilpotent bases for D. Moreover, the nilpotency orders of the generated real Lie algebras can be explicitly computed by means of simple linear algebra (for k = 1 this was done earlier in [M1], [M3]). For k = 2 we also transform our linear algebra formulas into recursive ones that resemble a bit Jean's formulas [Je] for nonholonomy degrees of Goursat germs.Additionally it is shown that, when all parameters appearing in a local form for a special k-flag vanish, then such a distribution germ is also strongly nilpotent in the sense of [AGau] and [M1].

Configurations of an Articulated Arm and Singularities of Special Multi-Flags

“…The notion of special multi-flags is described in [9,11]. Furthermore, for m ≥ 2, the existence of a completely integrable subdistribution F of D 1 implies property (iii).…”

“…A special multi-flag can be considered as a generalization of the notion of Goursat flags and the fundamental result of [2] and [14] is again obtained by Cartan prolongation (see also [9]). Consequently, in this situation, we can build a monster tower by successive Cartan prolongations of T R m+1 (see [2,3,4,14]):…”

“…Section 5 is devoted to the relation between EKR sets of depth 1 and RVT sets. In Section 5.1, we summarize the definition and the results concerning EKR classes based on [8,9]. Section 5.2 gives a global description of EKR sets in terms of RVT sets.…”

Orbital feedback linearization for multi‐input control systems

“…The following theorem, given by Pasillas‐Lépine and Respondek , see also , characterizes distributions, which are equivalent to the Cartan distribution . Theorem For a rank m + 1 distribution , with , on an open subset X of , the following conditions are equivalent: is, around any x 0 ∈ X , locally, equivalent to the Cartan distribution ; satisfies the following conditions: ; and , and ; if m = 2 then, additionally, the distribution is involutive; …”

“…The following theorem, given by Pasillas-Lépine and Respondek [21,22], see also [23][24][25], characterizes distributions, which are equivalent to the Cartan distribution C n .1; m/. (i) D is, around any x 0 2 X , locally, equivalent to the Cartan distribution C n .1; m/; (ii) D satisfies the following conditions:…”