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 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].
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