1. CR STRUCTURES

3

that (j) is both injective and surjective. Thus there are also

(N^k)

elements

in / . Since

(N+k)

=

(N£k),

this proves the lemma.

Let usfixsome hypersurface M given by v = F0(x, y, u), containing the

origin. Then any nearby surface is of the same form v = F(x, y, u). There

is a map from these hypersurfaces into

Rp,

p =

(iV^3)

, given by the TV-jet

at the origin of F. This map clearly covers a neighborhood of the origin.

Now let us consider the set 5? of local biholomorphisms of C defined in

a neighborhood of the origin and taking M to a hypersurface of the same

form. Thus we also have a map of S? into

Rp

. This map factors through

the map of S? into the space of TV-jets of pairs of holomorphic functions,

namely

Rq,

q =

4{N22)

. If M were equivalent to every other hypersurface

of the same form there would be a continuous map from R^ to

Rp

which

covers some neighborhood of the origin. But this is impossible whenever

T V 10 because in this case p q . This is our second proof that in general

hypersurfaces are inequivalent.

Now assume that points px and p2 are given on hypersurfaces M{ and

M2 and we ask what is the highest order of contact possible, in general,

between M2 and O(Mj) where O is a biholomorphism taking p{ to p2.

We may assume px — p2 = 0 and M{ and M2 both have the form v =

F(z, y, u). Then to make the TV-jet of O(Mj) agree with the TV-jet of

M2 involves (^

3

) - 1 conditions, while the TV-jet of 4 involves only

4{(N22)

- 1} parameters. Thus, as Poincare pointed out, one cannot achieve

ninth-order contact. In fact, as we shall see in Chapter 3, an obstruction

already is present to sixth-order contact.

These two simple arguments already point to our major topics. Since not

all hypersurfaces are locally equivalent it is natural to seek invariants which

allow us to distinguish one from another. The problem of finding these invari-

ants in

C2

was solved by Cartan [Ca2] and in a completely different manner

by Moser and then generalized to higher dimensions by Chern and Moser

[CM]. These two solutions to the problem posed by Poincare will occupy us

for most of this book. And the remainder can be traced to the observation

that a function on a hypersurface is the restriction of a holomorphic function

only if it satisfies a certain differential equation.

§2. CR manifolds. We start with some standard notation and concepts.

Let (Zj, ... , zn) be the usual coordinates for

Cn

and (x{ , y{, ... , xn , yn)

the corresponding coordinates for

R2"

. We define the first-order partial dif-

ferential operation

_d_ _ \_ (_d__ . _d_\

dZj-iyWj-'dyj)

and its conjugate operation

d_ _ \_ (_d_ . d \

Wj-iyaxj+'ifyj •