GEOMETRIC APPROACH TO GOURSAT FLAGS * Richard ...

Proof. Let E be the prolongation of the Goursat distribution D on M. The leavesof L(E 2 ) are the bers PD m of the bration : PD ! M, sothat M itself is canonicallyidentied with the leaf space PD=L(E 2 ). Now D = E 2 by the previous proposition,and D = D. This proves that the deprolongation of the prolongation is the original.Conversely, suppose that : U ! M is a local deprolongation, where E is the rank2 Goursat distribution on U, and D = (E 2 )isits deprolonged distribution. Writem = (u), with u 2 U. Thend u (E u ) D mis a one-dimensional subspace {an element ofPD m .Thus u ! d u (E u )denes a map:U ! PDfrom the original Goursat manifold to the prolongation PD of its (local) deprolongation.We claim that is a local dieomorphism. Indeed, is a ber bundle map over M, soallwe need to checkisthat the restriction of its dierential to L(E 2 ) u , the tangentspace to theber of : U ! M at u is onto. Moving along the leaf ` = ;1 (m) ofL(E 2 ) correspondsto owing with respect to a nonzero vector eld W 2 L(E 2 ). So we want toshow thatd u (W u ) 6= 0. Complete W to a local frame fWXg for E near u. Then [WX](u) 6= 0,mod E u since E 2 u 6= E u . This is equivalent tothe condition that d u (W u ) 6= 0. Finally,one easily checks that maps E to the prolongation of D. Q.E.D.5.4. Monster Goursat manifold. Proof of Theorem 1.Suppose that we had a Goursat distribution of corank s on a manifold M with thepropertythat every corank s Goursat germ was represented by some pointofthe manifold.Then the prolongation of M would enjoy the same property, but now among corank s +1Goursat distribution germs! For if we are given any corank s +1 Goursat distribution,its deprolongation is represented by somepoint ofM, byhypothesis. And by proposition5.3, upon prolonging this deprolongation we arriveatagerm dieomorphic to the original.There is such an M in the corank 2 case. Indeed, in this case, there is only onecorank 2, rank 2 Goursat germ up to dieomorphism. This is the Engel germ. Thus anyEngel distribution on a 4-manifold will serve forM, withs =2. It follows that everyGoursat germ of corank s +2 is realized within the s-fold prolongation of an Engeldistribution!Now anEngel distribution can be obtained by prolonging a contact structure on athree-manifold. And a contact three-manifold can be obtained by prolonging the tangentbundle to a surface (see the example of section 5.1). We have provedEvery corank s Goursat germ can be found, up to a dieomorphism, within the s-foldprolongation of the tangent bundle to a surface.We have called this s-fold prolongation the \monster manifold". It is a very tamemonster in many respects. Theorem 1 is proved.Remark. The direction of this section is in some sense opposite to that of sections3 and 4. In this section we imagine building Goursat distributions up from below by21