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442 PHILLIP A. GRIFFITHS We note that this connection is compatible with the metric, and as a consequence Now when n 2k is even, an n n skew-symmetric matrix A has a scalar invariant called the Pfaffian Pf(A), which has the property of being invariant under A BArB for B SO(n), and which satisfies Pf(A) det A. Since multiplication of forms of even degree is commutative, we may take the Pfaffian of the curvature matrix {lz}, and define (1.21) Pf(E) Cn J, A A 1,. lOl., A This is a closed invariant n-form on the Grassmannian, (11) and (for a suitable constant Cn) its de Rham cohomology class in the Euler class of the universal bundle. Returning to our even-dimensional oriented manifold Mn C IRN, the Gauss mapping y" M--) G(n, N) assigns to each x M the tangent plane Tx(M) Gn(n, N). Using Darboux frames, (1.4), and the PKicker embedding, the differential of the Gauss mapping is given by dy(x) d(ex A" A e.) i.e., the differential of the Gauss mapping is just the second fundamental form of M in IRN. Moreover, by definition 5’* E T(M), and we infer that the Riemannian connection on M is induced from the univer- (12) sal connection. As a consequence, y*() i and the Pfaffian form pulls back to A A,B C te I,,(RM) dM .BRoq(XdlB R.,, ,.,,._ lf. dM
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.
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