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496 PHILLIP A. GRIFFITHS
Footnotes
1. We recall that for V W a surjective proper mapping of complex manifolds with generic
fibre of dimension k, integration over the fibre gives a map
with the defining property
rr," Ac + k. + k)(V) Ac("q)(W)
(4.2) q/ rr*) zr.Ik/ /, 1 A*(W).
V W
It follows that 7r. commutes with 0 and c5, and also preserves positive forms. We shall also have
occasion to use integration over the fibre when V is singular, but the extension to these cases will be
obvious.
2. The point is that, by Schur’ lemma, any Hermitian form on s which is invariant under the
unitary group must be a multiple of the standard one. As much more sophisticated use of invariant
theory occurs in section 4(c) below--cf, the discussion pertaining to the reference cited in footnote
10.
3. cf. the reference given in footnote 5 of section 3.
4. cf. footnote 11 in section for an argument which covers all the cases we shall use in this
paper.
5. Once we know that, by averaging, the deRham cohomology of G(n, N) is isomorphic to the
cohomology
--
computed from the complex of invariant forms, the argument goes as follows: The
invariant forms are exterior polynomials in the quantities to,, bv which are invariant under
to, gto,vh,, where g and h are unitary matrices. Using the h’s, we deduce that only the
expressions
appear; in particular, the invariant forms are all of even degree. Using the g’s, any invariant form is
a polynomial in the elementary symmetric functions of (II). By direct integration one finds that
c(fE) is Poincar6 dual to
6. cf. W. Stoll, Invariant Forms on Grassmann Manifolds, Annals of Math. Studies no. 89,
Princeton University Press, (1977). The form r/F.a may also be constructed by applying standard
potential theory to the Grassmannian to solve the equation of currents (ii).
7. As we saw in (3.11), aside from a few special cases the Gaussian image y(M) may be expected
to have dimension n.
8. cf. footnote 2 in section 1.
9. We note that V T0(s) * N, and W Tn(G(n, N)) n N-n,; cf. footnote
3 in section 3.
10. H. Weyl, The Classical Groups, Princeton University Press, 1939.
11. In the reference cited in footnote 4) of section 2, cosets : and x are defined to be incident if gH
and g’K have in common a left coset relative to H tq K. This notion is not sufficiently broad to
cover the examples we have in mind.
12. cf. the recent works, Gary Jensen, Higher Order Contact of Submanifolds ofHomogeneous
Spaces, Lecture Notes in Math, no. 610, Springer-Verlag (1977); Mark Green, The moving frame,
differential invariants, and rigidity theorems for homogeneous spaces, to appear in Duke Math. J.
13. This means that C and C’ pass through x0 and have the same tangent line there.
5. Curvature and Piiicker Defects
(a) Gauss-Bonnet and the Pliicker paradox. To obtain an heuristic idea of