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
and dΦ brings
restricted to U to
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
level i + 1 defined by
Prolongation preserves the fibers of the fibration
of prolongation can be iterated. The k-step-prolongation
of Φ is the one-step-
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
(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
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].