16 1. CR STRUCTURES
and
U = bU (mod {L,L})
with A invertible and b real and nonzero. Then
g =
b~lAgA*
and so g has the same signature as g .
DEFINITION.
M2n+l
is nondegenerate at x if g(x) is a nonsingular ma-
trix.
In this case p + q = n .
DEFINITION.
Af2n+1
is strictly pseudoconvex at x if g(x) is either positive
definite or negative definite.
Here g has signature (w, 0) (or (0, «)) .
EXAMPLES.
2 and S are nondegenerate and Q(
q)
has signature (/?, q ,
n - p - f l ) .
REMARKS.
1. On M the concepts of nondegenerate and strictly pseudoconvex co-
incide. Further, in this dimension, they are equivalent to H being noninte-
grable.
3 3
2. To rephrase a previous observation: R and Q (or S ) are not locally
3 3
equivalent since Q (or S ) is everywhere nondegenerate and E is nowhere
nondegenerate.
Our main concern will be with nondegenerate submanifolds M c C .
We shall see that a great deal can be accomplished by modelling such hyper-
surfaces on the hyperquadric. Here is a first step.
LEMMA
4. If M is nondegenerate at p, then there is a complex affine
map srf which takes p to the origin and such that 3?(M) has the form
(15) v =
a\z\2
+
Az2 +l4~z2
+
bu2
+ Bzu + TlJu-\
with a ^ 0.
PROOF.
We first translate p to the origin. We then choose a point R on
HQ and a point S not on 7/0 and find the complex linear map which takes
R to (1,0) and S to (0, 1). We end up with a map sf{ so that s/{(H0)
is the z-axis. We now take a linear map of the form (z, w) (z,
£Z(^)
which takes
T Q J / ^ M )
to the (z,w)-plane. The composition map sf takes
Af to the form (15) and we need only show that a is nonzero. We see from
(14) that [L, L] has the form
[L,L]= L ( l - / 0 j | - + L(/0/ ^
9z
z
du
d
Tf
-M
\
.L(l + i ^ ) ^ - L ( - , ^ )
|S
-
and so, using (15), at the origin
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