Small growth vectors of the compactifications of the contact systems on J~r(1, 1)

摘要

It is well known that the compactifications of the canonical contact systems living on real jet spaces J~r(1,1), r ≥ 2, are locally universal Goursat distributions, Δ~r, living on compact manifolds (called Goursat monsters) having open dense jet-like (J~r(1,1)-like) parts. By virtue of the results of Jean (1996), one was able, for each r ≥ 2, to recursively compute the small growth vector of Δ~r at any point of the r-th monster. The result was got by performing series of r operations taken, in function of the local geometry of Δ~r in question, from the set of fixed recursive rules { 1, 2, 3 } in Jean's notation (called in the present text S, T, G, respectively). By the local universality of Δ~r one was thus able to compute all small growth vectors of all existing Goursat distributions. In the work of Mormul (2004) proposed were explicit solutions to the series of Jean's recurrences. Yet, those formulas appeared as if from thin air, and proofs were postponed to another publication. In the present contribution we submit proofs of the formulas from the year 2004. The paper is not a plain check that our candidates satisfy Jean's recurrences. We effectively retrieve the very formulas - they gradually (and naturally) emerge in the steps of an inductive proof.