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