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
Previous Page Next Page