GEOMETRIC APPROACH TO GOURSAT FLAGS * Richard ...

X t of vector elds on M. The relation X t cd! t = ; d! tin fact holds upon restriction todtthe entire space D t (p). This is because L(D t )(p) =kerd! t (p) and because the 1-form d! tdtvanishes on L(D t )(p). The latter fact is a consequence of the condition that L(D t )=L(D 0 )does not depend on t. This is the only place inthe proof where this condition is used.Now dene the path t of global dieomorphisms to be the solution to the ordinarydierential equation d t= Xdt t ( p )with the initial condition 0 = id. We will show that( t ) D t = D 0 . Wehave d (( dt t) ! t )= t (L Xt ! t + d! t), where L is the Lie derivativedtalong X t . Since X t is annihilated by ! t the Lie derivative isequaltoX t cd! t .Weshowedthat (X t cd! t + d! t)j dt D t (p) =0foranypoint p. This implies that X t cd! t + d! t= hdt t ! tfor some path of functions h t . Therefore the path of 1-forms A t =( t ) ! t satises thelinear ordinary dierential equation dA t= h ~ dt t A t with ~ h t = h t ( t )with initial conditionA 0 = ! 0 .We can integrate this equation. Indeed the ansatz A t = H t ! 0 yields the scalardierential equation dH t= ~ hdt t H t with solution H t =expf R t~h0 s dsg. Wehave shown thatA t := t ! t = H t ! 0 which means that ( t ) D t = D 0 .Q.E.D.Proof of Theorem A.2. Let ! be aglobal 1-form describing D, andlet ^! Nbeglobal 1-forms describing D N and such that ^! N ! ! in the Whitney C l+1 -topology. Sincethe (2r + 1)-forms ^! N ^ (d^! N ) r and ! ^ (d!) r have the same kernel L(D N )=L(D) ofcodimension 2r+1 then ^! N ^(d^! N ) r = H N !^(d!) r , where H N is a nonvanishing function.Replace ^! N by ! N = ^! NH r+1 : . The forms ! N also describe distributions D N , and we haveN! N ^ (d! N ) r = ! ^ (d!) r (A:1)The value of H N at any point depends on the values of ! ^! N and their dierentials at thesame point only, therefore H N ! 1inthe Whitney C l -topology. Consequently ! N ! !in the same topology.Dene the path! Nt = ! + t(! N ; !) t 2 [0 1]of one-forms. Let D Nt be the eld of kernels of ! Nt . Weshow that for suciently bigN the distribution D Nt is a corank one distribution of the same constant rank 2r +1and with the same characteristic foliation L(D Nt )=L(D) forall t 2 [0 1]. This followsimmediately from the following two statements:(a) ! Nt ^ (d! Nt ) r (p) 6= 0(forsciently big N, any t 2 [0 1], and any p 2 M)(b) d! Nt (Z Y Nt )=0for any vector eld Z 2 L(D) and any vector eld Y Nt 2 D Nt .Statement (a) follows from the C l -closeness of ! Nt to !, the compactness of the segment[0 1] and the condition l 1.To prove the second statement weuse the equality (A.1). Fix a vector eld Z 2 L(D).We know that Z(p) belongs to the kernel of d!(p)j D(p) for any point p of the manifold.This condition implies that Zcd! = h! for some function h. Similarly Zcd! N = h N ! Nfor some function h N . To prove (b) it suces to show that h N = h. Indeed, if h N = hthen for any vector eld Y Nt 2 D Nt wehave:28