GEOMETRIC APPROACH TO GOURSAT FLAGS * Richard ...

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The fact that d!j V 6=0implies that the rank of E 2 is greater than that of E. Butrank(E) =rank(D) ; 1 and E E 2 D. Consequently E 2 = D. Q.E.D.Consider the group Diff p of all local dieomorphisms with xed point p and itssubgroup Symm p (D) consisting of local symmetries of the germ at p of D:Symm p (D) =f 2 Diff p : D = D:gAny 2 Symm p (D) automatically preserve the canonical foliation L(D), and consequentlyit preserves L(D)(p). Its derivative d p thus acts on the two-dimensional factorspace D(p)=L(D)(p) byalinear transformation, and consequently denes a transformationThis denes a group homomorphismg : S 1 D(p) ! S 1 D(p) g :V = d p (V ) V 2 S 1 D(p): 7! g Symm p (D) ! PGl(2) = PGl(D(p)=L(D)(p)):We denote the image of this homomorphism by; p (D) =fg 2 Symm p (D)g:Remark: PGl(2) is the standard notation for the group of all invertible linear transformationsof a two-dimensional vector space modulo scale. Elements of this group maplines to lines, and hence dene transformations of RP 1 = S 1 . These transformations aresometimes called projectivities. So; p (D) isagroup of projectivities.Proposition 3.1 and Lemma 3.1 imply:Proposition 3.2. Let D be the germ at a point p of a Goursat distribution of coranks. Let E and ~ E be the germs at p of Goursat distributions of corank s +1 such thatE 2 = ~ E 2 = D. The germs E and ~ E are equivalent if and only if the points E(p) and ~ E(p)of the circle S 1 D (p) belong to a single orbit with respect to the action of the group ; p(D).The rest of this section is devoted to understanding the orbit structure of the actionof ; p (D) onthecircle.To understand the orbit structure we should rst understand the xed points of theaction. By a xed point V 2 S 1 D(p) wemeanapoint that is xed by every transformationin the group ; p (D). The set of all xed points will be denoted Fix p (D):Fix p (D) =fV 2 S 1 D (p) :g:V = V for any g 2 ; p(D)g:To reiterate V D(p) isacodimension 1 hyperplane which contains the codimension 2hyperplane L(D)(p), and g:V = d p (V )where g = g ,with2 Symm p (D).The set Fix p (D) isnever empty. Indeed, Symm p (D) preserves D 2 , and hence L(D 2 ).But L(D 2 )(p) D(p) isacodimension 1 hyperplane, as we sawinthe previous section(see the sandwich lemma). ConsequentlyL(D 2 )(p) 2 Fix p (D)10
for any Goursat distribution D.On the other hand, if Fix p (D) contain more than two points then Fix p (D) =S 1 D (p){ every point is a xed point, and ; p (D) =f1g consists of the identity transformationalone. This follows immediately from what is sometimes called \the fundamental theoremof projective geometry": any projectivity ofthe projective linewhich xes three or morepoints is the identity. At the level of linear algebra, this is the assertion that if a lineartransformation of the plane R 2 has three distinct eigenspaces (the three alleged xed pointsof the projective line ) then that transformation is a scalar multiple of the identity.We thus have the following possibilities. #(Fix p (D)) = 1, inwhich case ; p (D) =fidg, and the number ofinequivalentgerms E 2 p D is innite #(Fix p (D)) = 1, in which case that single xed point must be L(D 2 )(p) #(Fix p (D)) = 2, in which case the xed points are L(D 2 )(p) and one other point.The following proposition explores the middle possibilty.Proposition 3.3. If #(Fix p (D)) = 1 then Fix p (D) =fL(D 2 )(p)g.In this casethe action of ; p (D) is transitive away from the xed point. That is to say, for any twopoints V V ~ 2 S 1 (p) dierent from D L(D2 )(p) there existsag 2 ; p (D) such that g:V = V ~ .Consequently, the circle S 1 (p) consists of two orbits with respect to the group ; D p(D): thexed point L(D 2 )(p) and all other points.The proof of this proposition, and the one following (Proposition 3.4) are based onthe Lemma 3.2 immediately below. To appreciate the lemma, notice that the connectedpart of PGL(2) consists of projective transformations of the form exp(v) for some lineartransformation v of R 2 = D(p)=L(D)(p). Such alinear transformation can be viewed asa linear vector eld on the plane, and hence a vector eld v on the circle S 1 . (The vectorelds arising in this way are precisely the innitesimal projective transformations.) Theow exp(tv) ofthis vector eld is a one-parameter group of projectivities connecting theidentity toexp(v). The set of such v forms the Lie algebra of PGl(2), denoted pgl(2).Lemma 3.2.The square of; p (D) is connected. In other words, if g 2 ; p (D), theng 2 = g g = exp (v) for some vector eld v 2 pgl(2) on the circle S 1 D(p) with the propertythat exp (tv) 2 ; p (D) for all t 2 R.The proof of the Lemma is postponed to Appendix B.We will now investigate the case in which Fix p (D) consists of two points: L(D 2 )(p)and some V 6= L(D 2 )(p).Denition. If V 6= L(D 2 )(p), let = (D V p) : S 1 D (p) ! S1 D (p)denote the projectivity induced byareection in the plane D(p)=L(D)(p) whose xed pointset consists of the two points V and L(D 2 )(p) (mod L(D(p)).We explain. Let 2 RP 1 be two distinct points of the projective line. Choosecoordinates for the plane R 2 so that and are the x and y coordinate axis, and let11
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: If it were true that each Kumpera-R
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 and 28: the point p 2 M. The denition of th
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

GEOMETRIC APPROACH TO GOURSAT FLAGS * Richard ...

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