GEOMETRIC APPROACH TO GOURSAT FLAGS * Richard ...

R1. Let 2 Symm p (D). Then we can express 2 in the form 2 = 1exp(V ) wheretexp(tV ) 2 Symm p (D), V is a vector eld germ at p, vanishing at p and t is a familyof local dieomorphisms such that j 1 p t = id, t 2 R.Note that we are not asserting that t or exp(tV )liein Symm p (D).Step 2. Proof of (R1). Fix any k>s= corank(D). It follows from Proposition6.2, part 2 that if D ~ is a germ at p of a Goursat distribution such thatjkp D ~ = jkp D ~ thenthere exists a local dieomorphism such that D ~ = D and jk;sp =id. Inparticularjp=id. 1 Therefore to prove (R1)itsuces to prove the following statement:R2. Let 2 Symm p (D). Then there exists alocal vector eld V such that:j k+1p 2 = j k+1p exp(V ) (B:1)j k pexp(tV ) D = j k pD t 2 R: (B:2)Step 3. We show that (B.1) implies (B.2). It is clear that (B.1) implies (B.2) for allinteger t. ByProposition B.1, the eigenvalues of j 1 pare real, therefore the eigenvalues ofj 1 p 2 are positive and consequently those of j 1 V p are real. Therefore the relation (B.2) canbe expressed in the form F 1 (t) F m (t) 0, where each ofthe functions F 1 :::F mis a linear combination of real exponential functions with polynomial coecients. SinceF i (t) =0for any integer t then F i (t) 0 and (B.2) holds.Step 4. We have reduced Lemma 3.2 to the proof of the existence of a vector eld Vsatisfying (B.1). Let Jp k+1 be the space of the (k + 1)-jets at p of functions vanishing at p.Consider the linear operator A : Jp k+1 ! Jp k+1 such that A(f) =jp k+1 f( 2 ) f 2 Jp k+1 .To prove that (B.1) holds for some vector eld V it suces to show that the operator Aadmits a logarithm, i.e. that there exists a linear operator B : Jp k+1 ! Jpk+1 such thatA = exp(B). To show this it suces to prove that the eigenvalues of A are real positivenumbers. It is known that the eigenvalues of A have the form 11 nn ,where iare eigenvalues of the linearization of 2 at p, and where the i are non-negative integerswhich sum to k +1.ByProposition B.1 these i 's are real positive numbers. Therefore thesame is true for the eigenvalues of the operator A. The proof of Lemma 3.2 is completed.Proof of Proposition B.1. To prove Proposition B.1 we willshow that in suitablecoordinate system the matrix of j 1 p istriangular. Let D = D s D s;1 D 2 D 1be the Goursat ag generated by D. Take alocal coordinate system x 1 :::x n centeredat the point p such that the Engel subag D 2 D 1 is described by 1-forms! 1 = dx 1 ; x 2 dx 3 and ! 2 = dx 2 ; x 4 dx 3 and the characteristic foliations L(D i )havethe form (dx 1 :::dx i+2 ) ? , i =1:::s.Denote i =(x i ). The form of the characteristicfoliations and the fact that they are preserved by implies that @ i@x j(0) = 0 for j>iand j>3. To show that the matrix of the linear approximation of is triangular in thechosen coordinate system we have toprove that@ 1@x 2(0) = @ 1@x 3(0) = @ 2@x 3(0) = 0:(B:3)30