2 1. INTRODUCTION

are coordinates near p. A similar construction, yields coordinates on the higher

Monsters PiR2, i 1. See Chapter 7.

1.3. Symmetries. Equivalence of points of the Monster

A local symmetry of the Monster at level i is a local diffeomorphism Φ : U → U,

where U and U are open sets of

PiR2

and dΦ brings

∆i

restricted to U to

∆i

restricted to U.

Two points p, ˜ p of the Monster manifold PiR2 are equivalent if there exists a

local symmetry of PiR2 sending p to ˜. p

1.4. Prolonging symmetries

The prolongation of a local symmetry Φ at level i is the local symmetry

Φ1

at

level i + 1 defined by

Φ1(m,

l) =

(

Φ(m),dΦm( )

)

.

Prolongation preserves the fibers of the fibration

Pi+1R2

→

PiR2.

The process

of prolongation can be iterated. The k-step-prolongation

Φk

of Φ is the one-step-

prolongation of

Φk−1.

1.5. The basic theorem

Our whole approach hinges on the basic theorem:

Theorem 1.1. For i 1 every local symmetry at level i is the prolongation of

a symmetry at level i − 1.

Upon applying the theorem repeatedly, we eventually arrive at level 1, which

is the contact manifold

(P1R2, ∆1).

(See for example [A1] regarding this contact

manifold. See also section 1.2). The symmetries of a contact manifold are called

contact transformations, or contactomorphisms. Thus Theorem 1.1 asserts that

the (i − 1)-fold prolongation is an isomorphism between the pseudogroup of con-

tact transformations (level 1) and the pseudogroup of local symmetries at level i.

The theorem expressly excludes the isomorphism between the i = 0 and i = 1 pseu-

dogroups. Indeed the pseudogroup of contact transformations is strictly larger than

the first prolongation of the pseudogroup of local diffeomorphisms of the plane, see

[A1].

Theorem 1.1 can be deduced from our earlier work [MZ], namely from the

“sandwich lemma” for Goursat distribution and the theorem (Theorem 1.2 in the

next section) on realizing all Goursat distribution germs as points within the Mon-

ster tower. In order to be self-contained we present a simple proof of Theorem 1.1

in section 1.8 in purely Monster terms.

Remark. Our Monster construction started with 1-dimensional contact el-

ements for the plane. The construction generalized by starting instead with k-

dimensional contact elements on an n-manifold. The generalization of Theorem

1.1 remains valid, and holds even for i = 1 when k 1. This generalization is

sometimes called the Backlund theorem, and is tied up with the local symmetries

associated to the canonical distribution for jet spaces. See for example [B] and [Y].