GEOMETRIC APPROACH TO GOURSAT FLAGS * Richard ...

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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
d! Nt (Z Y Nt )=(1; t)d!(Z Y Nt )+td! N (Z Y Nt )==(1; t)h!(Y Nt )+th! N (Y Nt )=h! Nt (Y Nt )=0:To prove that h N = h we take the Lie derivative L Z of the relation (A.1) along thevector eld Z. Since Z belongs to the kernel of each ofthe (2r +1)-forms in (A.1), weobtain L Z (! ^ (d!) r )=Zc(d!) r+1 =(r +1)(d!) r ^ (Zcd!) =(r +1)h! ^ (d!) r and, inthe same way, L Z (! N ^ (d! N ) r )=(r +1)h N ! N ^ (d! N ) r . But (A.1) holds, and hence sodoes the Lie derivative of(A.1) with respect to Z. We conclude that h N = h.We have proved that the path of distributions D Nt satises the conditions of TheoremA.1. By this theorem there exists a dieomorphism N sending D N = D N1 to D = D N0 .Tracing the proof of Theorem A.1 we see that as N !1the dieomorphism N tendsto the identity dieomorphism in the same topology in which the 1-form d! Nttends todtzero 1-form. Since d! Nt= !dt N ; ! and ! N ! ! in the C l -Whitney topology, wehave that N ! id in the same topology. Q.E.D.We also need the following local version of Theorem A.2. Its proof is the almost thesame.Theorem A.3.Part 1 (for germs at a nonxed point). Let D and D N becorank one distributions ona manifold M of constant class 2r+1 described by1-forms ! and ! N such that j l p ~! N ! j l p !for some point p 2 M, and for l 1. Let U be any neighbourhood ofthe point p. Thenfor suciently large N there exist open sets (possibly disjoint) U N 1 U N 2 U with p 2 U N 1and a dieomorphism N : U1 N ! U2 N which sends the distribution D N restricted toU1Nto the distribution D restricted toU2 N ,andsatises j`;1p N ! j`;1p id as N !1.Part 2 (for germs at a xed point). Let D and ~ D be germs at a point p of corank onedistributions of constant class 2r +1 with the same l-jets at p, l 1. Then there exists alocal dieomorphism such that jp l;1 =jp l;1 id and D ~ = D.Note that in Part 1 in general N (p) 6= p. Tomakesense of the condition jp l;1 N !jpl;1 id one should take U to be a coordinate neighborhood and identify the `-th jet withthe `-th order Taylor expansion of N .Appendix B. Proof of Lemma 3.2.This lemma is based on the following statement.Proposition B.1. Let D be any Goursat distribution of corank s 2. All eigenvaluesof the linearization at p of any local symmetry 2 Symm p (D) are real.We prove this Proposition at the end of this Appendix. Toshowhowitimplies Lemma3.2 we need several reduction steps.Step 1. The projectivity g of the circle S 1 (D)(p) depends on jp 1 only. Thereforeto prove Lemma 3.2 it suces to prove the following statement:29
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 and 26: We willseek for H Nt in the form H
Page 27: the point p 2 M. The denition of th
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

GEOMETRIC APPROACH TO GOURSAT FLAGS * Richard ...

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