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CURVATURE AND COMPLEX SINGULARITIES 495
the Gauss-Bonnet integrand for M N L. It is important to note that Pf(fl c L)
is invariant under reversal of orientation of M C) L, and so the absolute value
sign in (4.45) only pertains to dL.
We have now evaluated as being the left hand side of (0.4). As for
/
IM II()[, it is clearly a second order invariant of M C IRN, and hence is expressed
as the integral in some universal polynomial of degree 2(n k) in the
second fundamental form of M. The main step is to show, by an argument using
Meusnier’s theorem, that this polynomial has the same invariance properties as
those in Weyl’s tube formula (1.10); the details are given in the reference cited
in footnote 6 of the introduction.
Example 4. The same procedure as in example 3 may be used in the complex
case, only the argument is simpler since we may take
without absolute value signs, and then
P c,_ (lv) / dL
I() (Trl).(Tr*)
is a G-invariant closed (n, n) form on 0(n, N), which we may seek to determine
without reference to any complex manifold M C u. This is the procedure followed
in our proof of (4.17).
Example 5. Here we take G U(N), G/K to be the manifold F(N) of all
flags Wo WI C WN N, and G/H the Grassmannian G(n, N). Then,
for a sequence a (al, ", a,) of integers as in section 4(a), we let
be defined by the Schubert conditions
I G(n, N) F(N)
I {(T, F): T a(F)}.
Taking dF the invariant volume on F(N) we have that
I() (Trl),(Tr*dF)
is an invariant form on G(n, N).
The determination of I() brings out one of the salient features of Hermitian
integral geometry. Namely, since the invariant differential forms on G(n, N)
are isomorphic to the cohomology, the form I() is uniquely specified by its
cohomology class, and consequently the determination of the mapping
I() is a purely topological question. In the case at hand
and this implies the formula (4.9).
I() P(f),