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CURVATURE AND COMPLEX SINGULARITIES 443
by (1.9). In other words, the Gauss-Bonnet integrand is induced from the universally
defined form Pf(I) under the Gauss mapping, and therefore in de
Rham cohomology it represents the Euler class of the tangent bundle, which
again implies the Gauss-Bonnet theorem for submanifolds of IR. One way to
prove the Gauss-Bonnet theorem for a general compact Riemannian manifold
M is to observe that the preceding argument applies provided only that the
Riemannian connection is induced from the universal connection by a map
f: M G(n, N) with f*E T(M); that such a classifying map exists follows
from the theorem of M. Narasimhan and Ramanan, Amer. J. Math., Vol. 83
(1961), pp. 563-572. An alternate proof which has worked well in teaching differential
geometry is given at the very end of this paper.
In closing we remark that the other terms in Hermann Weyl’s formula (1.10)
are not pullbacks of invariant forms an the Grassmannian, and (as previously
noted) are of a metric rather than a topological character. We are perhaps belaboring
this point, because the complex case will be in sharp contrast in that all
the terms turn out to be rigid.
Footnotes
1. H. Weyl, On the volume of tubes, Amer. J. of Math., vol. 61 (1939), pp. 461-472; cf. also H.
Hotelling, Tubes and spheres in n-spaces and a class of statistical problems, American J. Math.,
vol. 61 (1939), pp. 440-460.
2. A differential system on a manifold X is given by an ideal in the algebra of differential forms
which is closed under d; an integral manifold is a submanifold Y C X such that b[y 0 for all
3. Recall that
so that the normal part of
dx 2 toeandde= Z to13 e13 + Z oo,e,,
d2x is
4. H. Flanders, Development of an extended differential calculus, Trans. Amer. Math. Soc.,
vol. 57 (1951), pp. 311-326.
5. cf. C. B. Allendoerfer, The Euler number ofa Riemannian manifold, Amer. J. Math., vol. 62
(1940), pp. 243-248; and C. B. Allendoerfer and A. Weil, The Gauss-Bonnet theorem for Riemannian
polyhedra, Trans. Amer. Math. Soc., vol. 53 (1943), pp. 101-120. When n 2 so that M is a
surface, we have In(RM) CeR1212, which implies that/z2(M) is a constant times the classical Gauss-
Bonnet integral
6. Recall that the degree is the number of points, counted with --. signs according to orientations,
in y-l(e) for a general value e S s- 1. This result is an immediate consequence of Hopf’s
theorem about the number of zeros of a vector field (cf. H. Hopf, Vektorfelder in Mannigfaltigkeiten,
Math. Ann. vol. 96 (1927), pp. 225-250.)
7. Actually, this program will be discussed in some detail at the very end of the paper---cf.
section 5d.