CHAPTER 1
Introduction
1.1. The Monster construction
The Monster is a sequence of circle bundle projections
. . .
Pi+1R2

PiR2
. . .
P2R2

P1R2

P0R2
=
R2
between manifolds
PiR2
of dimension i + 2, i = 0, 1, 2,... , each endowed with a
rank 2 distribution
∆i
T
PiR2.
The construction of
(PiR2, ∆i)
is inductive.
(1)
P0R2
=
R2, ∆0
= T
R2.
(2)
Pi+1R2
is the following circle bundle over
PiR2:
a point of
Pi+1R2
is a pair
(m, ), where m
PiR2
and is a 1-dimensional subspace of the plane
∆i(m).
The rank 2-distribution
∆i+1
is defined in terms of smooth curves tangent to
∆i+1.
A smooth γ : (a, b)
PiR2
is said to be tangent to
∆i,
or integral, if
γ (t)
∆i
(curve
γ(t)
)
, t (a, b).
(3) a curve t
(
m(t), (t)
)
in
Pi+1R2
is tangent to
∆i+1
if the curve t m(t) in
PiR2 is tangent to ∆i, and m (t) (t) for all t.
The construction of Pi+1R2 from PiR2 is an instance of a general procedure
called prolongation due to E. Cartan [C1, C2, C3] and beautifully explained in
section 3 of [BH]. (Cartan invented several procedures now known as “prolon-
gation”. At a coordinate level, these prolongation procedures involve extending
previously defined objects by adding derivatives.) The description we have given
is repeated from our earlier work [MZ]. The symbol P is used to denote projec-
tivization :
Pi+1R2
is the projectivization
P∆i
of the rank 2 vector bundle
∆i
over
P
iR2.
When we refer to the Monster at level i, or the ith level of the Monster, we
mean
PiR2
endowed with the distribution
∆i.
Convention. Throughout the paper, all objects (diffeomorphisms, curves,
etc.) are assumed to be real-analytic.
1.2. Coordinates and the contact case
The first level of the Monster,
P1R2,
is a well-known contact 3-manifold. See
for example [A1]. It is the space of lines in the plane, and is diffeomorphic to
R2
×
RP1.
A point p
P1R2
is a pair (m, ), m
R2
and is a line in the
tangent plane
TmR2.
Standard local coordinates near p are (x, y, u), where (x, y)
are Cartesian coordinates for
R2
and u is an affine coordinate on the fiber
RP1
of
P1R2 = R2 × RP1. To construct u, suppose that is not parallel to the y-axes:
dx| = 0 and set u = dy(v)/dx(v), where v is any vector spanning . In other words,
u is the slope of the line and has the “hidden” meaning of dy/dx. Rearranging
u = dy/dx we have dy udx = 0 which defines the contact structure ∆1 near
p. Relative to these coordinates the projection P1R2 R2 is (x, y, u) (x, y).
If is parallel to the y-axis then we use instead ˜ u = dx(v)/dy(v) and (x, y, ˜) u
1
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