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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