GEOMETRIC APPROACH TO GOURSAT FLAGS * Richard ...

Remarks.1. Note that in part 1 wemayhave N (p) 6= p for all N. Tomakesense of the conditionj l;1p N ! j l;1pid one should take U to be a coordinate neighborhood and identify the `-thjet with the `-th order Taylor expansion of N .2. Proposition 6.1 and the rst part of Proposition 6.2 hold for l 1whereas thesecond part of Proposition 6.2 also covers the case l =0. Thisdierence is essential. Thecase l =0is necessary for the proof of Proposition 3.1 and the proof of s-determinacy inTheorem 2.Proof of Proposition 3.1. This is the case l =0of Proposition 6.2, part 2.Proof of Theorem 3. Let F : D s D s;1 D 2 D 1 and ~ F : ~ Ds ~D s;1 ~ D2 ~ D1 be Goursat ags on manifold M described by C s+1 -close tuples! 1 :::! s and ~! 1 :::~! s of 1-forms. Assume that the foliations L(D 1 ) and L( ~ D 1 ) arethe same. By Theorem A.2 (Appendix A) there exists a C s -close to the identity dieomorphism 1 of M which brings D ~ 1 to D 1 . This dieomorphism brings the ag F ~ tothe ag ( 1 ) F ~ : (1 ) Ds ~ ( 1 ) Ds;1 ~ ( 1 ) D2 ~ D 1 described by thetupleof 1-forms ! 1 1 ~! 2::: 1 ~! s which isC s;1 -close to the tuple ! 1 ! 2 :::! s . Now weapply Proposition 6.1 with s =2there and the ` there equal to the current s ; 1. Itguarantees the existence of a C s;1 -small dieomorphism 2 which bringsthe length 2 ag( 1 ) D2 ~ D 1 to the ag D 2 D 1 . This dieomorphism brings the ag ( 1 ) F ~ to theag ( 2 1 ) F ~ : (2 1 ) Ds ~ ( 2 1 ) Ds;1 ~ ( 2 1 ) D3 ~ D 2 D 1 describedby the tuple of 1-forms ! 1 ! 2 ( 2 1 ) ~! 3 :::( 2 1 ) ~! s which isC s;2 -close to the tuple! 1 ! 2 ! 3 :::! s . Continue applying Proposition 6.1 (s ; 3) times more we toobtaina sequence of dieomorphisms 3 ::: s;1 for which thecomposition s;1 s;2 1brings the ag F ~ to the ag ^F described by 1-forms !1 ! 2 :::! s;1 ^! s ,where^! s =( s;1 s;2 1 ) ~! s . The 1-forms ^! s and ! s are C 1 -close. Using Proposition 6.1 forone last time we obtain a dieomorphism s which brings the ag ^F to the ag F . Thedieomorphism s s;1 s;2 1 brings the ag F ~ to the ag F . Q.E.D.Proof of Theorem 2 { structural stability. This follows from Theorem A.3 part1 and the Proposition 6.2 part 1 in the same way that Theorem 3 followed from TheoremA.2 and Proposition 6.1.Proof of Theorem 2{ s-determinacy.The proof is essentially the same as the proof of theorem 3 above, except we useTheorem A.3, part 2 instead of theorem A.2, and the second part of Proposition 6.2 insteadof Proposition 6.1. Namely, we start with two germs F and ~ F at a xed point p of Goursatags of length s described by s-tuples of 1-forms ! 1 :::! s;1 ! s and ~! 1 :::~! s;1 ~! s asin the proof of theorem 3 above, and having the same s-jets at p. Using Theorem A.3,part 2 and then Proposition 6.2, part 2, s ; 2times we conclude that ~ F is equivalent tothe germ of another Goursat ag ^F at p, where ^F is described by thetuple of 1-forms! 1 :::! s;1 ^! s and where ^! s (p) =!(p). Now apply Proposition 6.2, part 2, with l =0to conclude that the germ of ^F is equivalent tothe germ of F .23