GEOMETRIC APPROACH TO GOURSAT FLAGS * Richard ...

the point p 2 M. The denition of the class is due to [Frobenius, 1887] and [Cartan, 1899].For example, the class of a contact structure is the dimension of the underlying manifold.The maximal possible class of a corank one distribution on a manifold of evendimension 2k is 2k ; 1. Such adistribution is called a quasi-contact , or even-contact,structure. A foliation of codimension one has class 1, the minimal possible class. In section2 we proved that the corank one distribution D 1 of a Goursat ag has constant class3.Recall that the characteristic foliation L(D) ofthe distribution D is the foliationgenerated by vector elds X 2 D such that [X D] D, i.e. [X Y ] 2 D for any Y 2 D.The characteristic foliation L(D) D for a corank 1 distribution D of constant class 2r+1has codimension 2r within D. Itisthe kernel of the 2-form d!j D(p) ,where ! is as above.(See the proof of Lemma 2.2.) This kernel coincides with the kernel of the (2r + 1)-form! ^ (d!) r (p) onthe space T p M. (By the kernel of an exterior q-form on a vector spacewe mean the subspace of vectors v such that the form annihilates every q-tuple of vectorscontaining v.)For example, the characteristic foliation of a quasi-contact structure is a line eld.The characteristic foliation of a contact structure is trivial: it is the zero section of thetangent bundle. The characteristic foliation of an involutive corank one distribution is thedistribution itself. The characteristic foliation of the corank one distribution of a Goursatag has codimension 3 within the manifold.The following theorems generalizes Gray's theorem. By a cooriented corank one distributionwe mean a distribution which canbeglobally described by a1-form.Theorem A.1. Let D t beapath of cooriented corank one distributions on a compactmanifold M of constant class 2r +1 such that L(D t )=L(D 0 )t2 [0 1]. Then there existsapath t of global dieomorphisms of M such that ( t ) D t = D 0 t 2 [0 1].For quasi-contact structures Theorem A.1 is known to specialists, although is unpublishedto our knowledge. ,Using Theorem A.1 we obtain Theorem A.2 below. We needitforour proofs ofTheorems 2 and 3 in the body of the present paper, where it is applied to the case ofcorank one distributions of constant class 3.Theorem A.2. Let D and D N N =1 2::: be cooriented corank one distributionson a compact manifold M of constant class 2r +1such that D N ! D as N !1in theC l+1 -Whitney topology, l 1, and L(D N )=L(D) for all N. Then there exists a sequence N of global dieomorphisms of M such that N ! id as N !1in the C l -Whitneytopology and ( N ) D N = D 0 for suciently big N.Proof of Theorem A.1. Fix a Riemannian structure on M. For p 2 M, denote byV t (p) D t (p) the 2r-dimensional subspace of D t (p) whichisthe orthogonal complementto L(D t )(p) with respect to this metric. Let ! t be the path of 1-forms describing D t .The form 2-form d! t j Vt (p) is nondegenerate because L(D t )=ker d! t j Dt (p). Therefore theequation (X t (p)cd! t )j Vt (p) = t (p) has a unique solution X t (p) 2 V t (p) for any 1-form t (p) onV t (p). We need this solution when t = ; d! tj dt V t (p): The solution X t (p) dependssmoothly (analytically) on the point p and on t, and so denes a smooth (analytic) path27