1.6. THE MONSTER AND GOURSAT DISTRIBUTIONS 3

1.6. The Monster and Goursat distributions

Our earlier work [MZ], where the Monster was introduced, was motivated by

the problem of classifying Goursat distributions. Given a distribution D ⊂ TM

on a manifold M we can form its “square” D2 = [D, D], where [·, ·] denotes Lie

bracket. Iterate, forming Dj+1 = [Dj,Dj]. The distribution is called “Goursat” if

the Dj have constant rank, and this rank increase by one at each step rank(Dj+1) =

1 + rank(Dj), up until the final step j at which point Dj = TM.

Theorem 1.2 ([MZ]).

(1) The distribution

∆i

on

PiR2

is Goursat for i ≥ 1.

(2) Any germ of any rank 2 Goursat distribution on a (2+i)-dimensional manifold

appears somewhere in the Monster manifold

(PiR2, ∆i):

this germ is diffeomorphic

to the germ of ∆i at some point of PiR2.

This theorem asserts that the problems of classifying points of the Monster and

of classifying germs of Goursat 2-distributions are the same problem.

1.6.1. Darboux, Engel, and Cartan theorems in Monster terms. A

rank 2 Goursat distribution on a 3-manifold is a contact structure. A rank 2 Goursat

distribution on a 4-manifold is called an Engel structure. Classical theorems of

Darboux and Engel assert that all contact structures are locally diffeomorphic and

that all Engel structures are locally diffeomorphic. (See, for example [A1], [VG],

[Z3]). In Monster terms:

Theorem 1.3 (Darboux and Engel theorems in Monster terms). All points of

P1R2

are equivalent. All points of

P2R2

are equivalent.

For i ≥ 2 not all points of PiR2 are equivalent, but there is a single open dense

equivalence class. Cartan found the normal form for the points of this class, i.e. he

wrote down the generic Goursat germ. In Monster terms:

Theorem 1.4 (Cartan theorem in Monster terms). There is a single equiva-

lence class of points in PiR2 which is open and dense. The germ of the 2-distribution

∆i at any point of this class is diffeomorphic to the 2-distribution described in co-

ordinates (x, y, u1,...,ui) by the vanishing of the 1-forms

dy − u1dx, du1 − u2dx, du2 − u3dx, · · · , dui−1 − uidx.

1.6.2. Some history. Cartan [C1], [C2], [C3] asserts a version of Theorem

1.2. However, Cartan defines prolongation by taking the usual derivatives of coor-

dinates, which is to say, his prolongation is aﬃne and does not allow tangent lines

to “go vertical”. When interpreted in this aﬃne sense, Cartan’s assertions are,

apparently, the (false) theorem that all Goursat distributions in dimension k + 2

are locally diffeomorphic to the distribution of Theorem 1.4. Indeed, in his famous

five-variables paper, [C4] Cartan seems to assert that the only Goursat germ is the

open and dense one. In [GKR] the authors found a counterexample to exactly this

assertion, and this example is the first Goursat singularity. Goursat in his book

[Go] presented the assertion of Theorem 1.4. A number of decades later Bryant

and Hsu [BH] redefined Cartan’s prolongation in projective terms, which is the

prolongation we have just used in defining the Monster. For more on the history,

see the introduction to [GKR] and section 3.1 of [BH].