GEOMETRIC APPROACH TO GOURSAT FLAGS * Richard ...

1. INTRODUCTION AND MAIN RESULTSThis paper is devoted to Goursat distributions and Goursat ags. A Goursat ag oflength s on a manifold M n of dimension n 4isachainD s D s;1 D 3 D 2 D 1 D 0 = TM s 2 (F )of distributions on M n (subbundles of the tangent bundle TM n of constant rank) satisfyingthe following (Goursat) conditions:corank D i = ii =1 2:::sD i;1 = D 2 i whereD2 i := [D iD i ] i =1 2 s: (G)The rst condition means that D i (p) isasubspace of T p M n of codimension i, forany pointp 2 M n . It follows that D i+1 (p) isahyperplane in D i (p), for any i =0 1 2:::s; 1and p 2 M n .Incondition (G) we use the standard notation D 2 or [D D] for the sheaf ofvector elds generated by D and the Lie brackets [X Y ], X Y 2 D, ofvector elds in D.By a Goursat distribution we mean any distribution of any corank s 2 of any Goursatag (F).An equivalent denition is as follows. Adistribution D of corank s 2isGoursat ifthe subsheaves D i of the tangent bundle dened inductively by D i+1 =[D i D i ](i =1 2:::s D 1 = D) correspond to distributions, i.e. they have constant rank, and thisrank is rank D i+1 = rank D i +1i=1:::s:Since the whole ag (F) is uniquely determined by the distribution D = D s of thelargest corank, we will say that D = D s generates (F). The study of Goursat ags andGoursat distributions is the same problem.The name \Goursat distributions" is related to the work [Goursat, 1923] in whichGoursat popularized these distributions. Goursat's predecessors were Engel and Cartan.Engel studied the case n =4s=2. This is the only case where the Goursat conditionholds for generic germs. He proved [Engel, 1889] that the germ of such adistribution isequivalent toasingle normal form without parameters. (See (C) below.)If (n s) 6= (4 2) then the set of germs of Goursat distributions of corank s on M n is asubset of innite codimension in the space of all germs. Nevertheless, Goursat distributionsappear naturally through Cartan's prolongation procedure. See, for example, [Bryant,1991] and section 5 of the present paper. The simplest realization of prolongation leads toa canonical Goursat 2-distribution (i.e., distribution of rank 2) on the (2 + s)-dimensionalspace of s-jets of functions f(x) inone variable. This distribution can be described by sdierential 1-forms! 1 = dy ; z 1 dx ! 2 = dz 1 ; z 2 dx : :: ! s = dz s;1 ; z s dx (C)where y represents the value of f at x and z i represents the value at x of the i-th derivativeof f. Cartan proved that ageneric germ of a Goursat 2-distribution can always bedescribed by the 1-forms (C). Indeed he proved the stronger statement:2