GEOMETRIC APPROACH TO GOURSAT FLAGS * Richard ...

Proof of Proposition 6.1. The proof will consist of three steps.First step. We will show that for suciently large N the agF Nt : D sNt D s;1 D 1 D sNt =(D s ! Nt ) ! Nt = ! + t(! N ; !)is a Goursat ag for any t 2 [0 1]. To show this we have tocheck the following statements:N(a) ! Nt j DsNt (p) is a nonzero 1-form for any p 2 M, t 2 [0 1] and suciently large N(b) d! Nt j DsNt (p) is a nonzero 2-form for any p 2 M, t 2 [0 1] and suciently large(c) if is a 1-form annihilating the distribution D s;1 then dj DsNt (p) =0forany N,any p 2 M and t 2 [0 1].The statements (a) and (b) follow from the fact that they are valid for t =0,thecondition that ! N tends to ! in the C 1 -Whitney topology (here we usethat l 1intheformulation of Proposition 6.1), and the observation that the hyperplane D sNt (p) aswellas the restrictions of the forms ! Nt and d! Nt to this hyperplane depend on the 1-jet atp of the form ! Nt only.To prove(c)weconsider the space L(D s;1 )(p). By the sandwichlemma 2.1 it is a codimension2 subspace of D s;1 (p) and the 1-forms ! and ! N annihilate this space. Therefore! Nt annihilates L(D s;1 )(p) for all t, i.e. L(D s;1 )(p) isahyperplane in D sNt (p), independentofN. Because L(D s;1 )(p) isthekernel of the 2-form d restricted to D s;1 (p), where annihilates D s;1 but not D s;2 (see Lemma 2.2), any hyperplane in D s;1 (p) containingL(D s;1 )(p) isisotropic for d. Inparticular, D sNt (p) isisotropic for d.Second step. We have proved that F Nt is a Goursat ag for suciently large Nand all t 2 [0 1]. In what follows assume that N is suciently large. Now we start toconstruct a path Nt of global dieomorphisms such that ( Nt ) F Nt = F N0 = F and inparticular ( N1 ) F N = F . Weuse the homotopy method. The second step of the proofis to reduce the construction of Nt to the construction of a path X Nt of global vectorelds satisfying the linear equations(X Nt cd! Nt + ! N ; !)j DsNt =0 X Nt 2 L(D s;1 ): (6:1)Assume that X Nt satises (6.1). Consider the following ordinary dierential equation andthe initial condition with a parameter p 2 M:d Nt (p)dt= X Nt ( Nt (p)) N0 (p) =p p 2 M: (6:2)Since M is a compact manifold and t varies on the compact segment [0 1], the solutionof (6.2) is a path Nt of global dieomorphisms on M. Let us show that ( Nt ) F Nt =F N0 . The condition X Nt 2 L(D s;1 )impliesthat Nt preserves the distribution D s;1 .Therefore to show that ( Nt ) F t = F 0 it is suces to show that there exists a path H Ntof nonvanishing functions such that(H Nt Nt! Nt ; ! 0 )j Ds;1 0: (6:3)24