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482 PHILLIP A. GRIFFITHS
unable to establish (4.21). As will seen below, (4.17) would be an immediate
consequence of this result.
What we will do is prove enough of (4.21) for our purposes. Namely, on
u G(n, N) there is an intrinsic differential ideal 8 # defined as follows: Over
{z, T} we consider frames {z; el," ", eN} where T-z el/k. /k en. Then the
forms
o, do, (/x n + 1,.. ", N)
generate an intrinsic differential ideal , and we shall prove that
(4.22) c,_ () modulo #.
Assuming this result for a moment, we will complete the proof of (4.17). For
this consider M as embedded in (n, N) n x G(n, N) by the refined Gauss
mapping
z {z, Tz(M)}.
Using Darboux frames we see that M is an integral manifold of the differential
system #, so that by (4.22) and (4.2) (cf. footnotez)
( c"-(u) *
f_ I(M
((N-k,N)( fMnL Cn-k(u))dL.
Now M L is mapped into (n k, N
generic L, and so
M 7) L
k, N) by z {z, Tz(M L), L} for
Cn k(U) (M L Cn k(M L)"
Combining with the previous step gives (4.17).
Turning to the proof of (4.22), we first note that the form $ on N
is invariant under the group of transformations
G(n, N)
{z, T} {gz + b, g T}
where g U(N) and b u. It follows that O is determined by its value at one
point, say {0, "}. If we write the (1, 0) cotangent space at this point as
V W,