GEOMETRIC APPROACH TO GOURSAT FLAGS * Richard ...

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By the sandwich lemma L(D s;1 )(p) isahyperplane in D sNt . Every such hyperplane isisotropic, so (6.6) implies that either X pNt 2 L(D s;1 )(p) orthat X pNt is a nonzero vectorin the kernel of the 2-form (d! Nt )j DsNt . The latter possibility isexcluded by the conditionX pNt 2 V Nt (p), the orthogonal complement toL(D sNt )(p) =ker(d! Nt )j DsNt .Proposition 6.1 is now proved. Q.E.D.Proof of Proposition 6.2. The proofs of the statements of Proposition 6.2 with`>0 are almost the same as as the proof we have justgiven. The dierence occurs mainlyin the construction of the dieomorphism Nt by the ordinary dierential equation (6.2).Concerning the case of part 1, the problem is that if X Nt is a time-dependent vector eldson a neighborhood U of a point p then its ow willtypically map out of that neighborhood{ hence the business with domains U N i in part 1. Although there may benosingle ow Nt t2 [0 1] of dieomorphisms on a single neighborhood of p, nevertheless, for N largethe vector X Nt (p) issuciently close to zero so that its solution denes dieomorphisms Nt : U N 1 ! U N 2t, t 2 [0 1], where U N 1 is a neighbourhood of p contained in U and U N 2t isan open subset of U (which mayormay not contain p).In the case of part 2 we have toshowthat t (p) =p and U 2t contains p. This followsbecause X Nt (p) =0for all t.The proof of Proposition 6.2 part 2 with l =0is also the same, except that we meet adicultyinthe rst step of the proof. Wehavetoshow that the restriction t (p) ofthe formd!+t(d~!;d!)tothe space D s (p) = D ~ s (p) doesnot vanish for all t 2 [0 1]. This is true fort =0and t =1,but if l =0then d!(p) might not be close to d~!(p) even in the C 0 -topologyand consequently t (p) mightvanish for some t 2 (0 1). Since t (p) depends linearly on t,this is impossible if 0 and 1 dene the same orientation of the 2-space D s (p)=L(D s )(p)(the orientations are well-dened since L(D s )(p) isthe kernel of 0 (p) and 1 (p)). If theorientations are dierent then we have toshow the existence of a symmetry of the germat p of the distribution D s;1 which also preserves !(p) and the foliation L(D s )(p) andchanges the dened above orientation. We can nd local coordinates centered at p suchthat L(D s;1 )=(dx 1 :::dx s+1 ) ? and L(D s )=(dx 1 :::dx s+1 dx s+2 ) ? , and such thatthe forms dening the D i can be taken to be independent ofx j , j s +2. It follows fromthe sandwich lemma that the dieomorphism x s+2 !;x s+2 is a symmetry of the requiredtype.Appendix A. Generalization of the Gray TheoremGray's theorem states that for any path of global contact structures D t t2 [0 1] on anodd-dimensional manifold M there exists a family of global dieomorphisms t : M ! Msuch that ( t ) D t = D 0 t 2 [0 1]. See [Gray, 1959]. It follows that two global contactstructures D and ~ D are equivalentprovided that ~ D is suciently close to D in the WhitneyC 1 -topology.In this section we generalize Gray's theorem to corank one distributions D of anyconstant class. Let ! be any nonvanishing 1-form describing D near p. Bytheclass of Dat p we will mean the odd number 2r+1 such that !^(d!) r (p) 6= 0and!^(d!) r+1 (p) =0.The even integer 2r is the rank of the restriction of the two-form d! p to D p .A corank one distribution has constant class if this class 2r +1does not depend on26
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
Page 1 and 2: GEOMETRIC APPROACH TO GOURSAT FLAGS
Page 4 and 5: Theorem 1. Apply the Cartan prolong
Page 7 and 8: Lemma 2.2. Let (F) be the Goursat a
Page 9 and 10: If it were true that each Kumpera-R
Page 11: for any Goursat distribution D.On t
Page 14: transformation and a single reectio
Page 17 and 18: (B 1 D 1 E 1 ) D 5 (0) 6= L(D 3 )(0
Page 19 and 20: Returning to the general rank 2 pro
Page 21 and 22: Proof. Let E be the prolongation of
Page 23 and 24: Remarks.1. Note that in part 1 wema
Page 25: We willseek for H Nt in the form H
Page 29 and 30: d! Nt (Z Y Nt )=(1; t)d!(Z Y Nt )+t
Page 31 and 32: To prove (B.3) we use the relations
Page 33 and 34: and the local classication of arbit
Page 35 and 36: References1887 Frobenius F., Uber d

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