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438 PHILLIP A. GRIFFITHS The point is that the blank area A is congruent to the doubly shaded region B. A second remark concerns a pair of Riemannian manifolds M1 and M2 with M1 Mg being given the product metric. Denoting by RM, + Me. /2"2(T(Ma x the curvature operator introduced in the formulation of (1.10), with the fairly obvious notation we have RM x M RM + RM which implies that /l RM x M2 /PRM + p RMz. Taking traces and integrating gives the functoriality propey (1.14) (M x M) (M) (Me). =0 This together with the reproductive property (0.4) possibly serve to charactere the curvature integrals (M). The final observation is that, even ifM is a compact mangold without boundary, the (M) are metric but not in general topological invariants. For example, for small r the dominant term in Weyl’s formula is, as expected, C vol (M)r. On the other hand, the coecient (IM, n 0(, of the highest power of r does turn out to be topologically invariant. This realization was intimately connected with the discovery of the higher dimensional Gauss-Bonnet theorem, and we should like to briefly recount this development. ( The starting point is the following theorem of H. Hopf: For H C N a compact oriented hypersufface, we consider the Gauss map " H S- sending each point y H to the outward unit normal p(y). HopFs result is that the degree of this map is a constant times the Euler-Poincar characteristic x(H). ( Expressed in terms of integrals (1.15) d C x(H) .f where d is * (volume form on S ). For example, using Darboux frames {y; e, ., e_ ; e} associated to H, we have
so that CURVATURE AND COMPLEX SINGULARITIES 439 v(y) eN, dv O)N,o e. E (.O(x,N e. E hahN 00 e where II h.NOo (R) eN is the second fundamental form of H. Thus do-= / tON,. Ct- KdH where K is the Gauss-Kronecker curvature (1.8), and so by Hopf’ s theorem we infer that .In Ct- KdH x(H), which is the Gauss-Bonnet theorem in this case. Using tubes we may pass from a general Mn C IR u to a hypersurface. Namely, assuming that M is oriented and compact without boundary, for sufficiently small r0 the boundary of the tube 0(M) will be an oriented hypersurface H; for simplicity of notation let us assume that r0 1. Expressing points y H in the form y x + ’, t, e,, Iltll 1, as in the proof of Weyl’s formula, the Gauss mapping on H is given by v(y) t. e,. By the structure equations (1.16) dr(y) ( . t..)e + . (dt. + t.)e.. The volume form on the unit normal sphere at x is C (-1)"-t. dt+A.. Ad&A..Adts. Using t. dr. 0 on H, it follows from (1.16) that
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For k we have (4.33) CURVATURE AND
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As a consequence of (5.15), CURVATU
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