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452 PHILLIP A. GRIFFITHS
as follows: the left hand side of (2.7) may first of all be given as an integral over
M of an expression in the second fundamental form of M. Secondly, since the
left hand side is invariant under isometric deformations ofM (i.e., bending), the
same will be true of the expression in the second fundamental form, which must
then be a polynomial in the curvature RM of M.
Of course this is not a proof, but it can be developed into one, as will be done
in 4(c) for the complex case. The point we should like to emphasize is that the
polynomials I,_ k(RM) are forced onto us in response to the question about
expressing the average (2.6) as an integral over M, and there they turn out to be
the same as those appearing in the formula for the volume of tubes. Unfortunately
we do not know a direct way of establishing a link between these two. (1)
A final remark is that the general reproductive property (0.4) may be deduced
from the special case (2.7) when n k dim (M N L), this in the obvious
way by expressing the left hand side of (0.4) as an interated integral of
/zk (M f-I L C) L’)’s where dim (M N L C) L’) k.
The formula (2.7) is a special case of the general kinematic formula of Chern
and Federer (cf. the references in footnote (6) of the introduction). This deals
with the following question: Given a pair of manifolds Mi (i 1, 2) of dimensions
ni with n + n2 -> N, as g E(N) varies over the Euclidean group, express
the average of the Euler characteristics
M1 n gM2)dg
in terms of invariants of the M. Here dg is the kinematic density, i.e., the biinvariant
measure on E(N). Formulas of this general sort originated with
Blaschke (cf. the references cited in (5) of this section) and are among the deepest
results in integral geometry. Their extension to the complex case in being
done by Ted Shifrin in his Berkeley thesis.
A final remark is this: On M, C IR N the Gauss-Bonnet integrand In(RI)dM is
invariant under reversal of orientation. This is a reflection of the topological
invariance of the Euler characteristic, and is proved by noting that both the
Pfaffian Pf(R) and volume form dM change sign under reversal of orientation,
so their product remains invariant. On the other hand the Pontrjagin forms
change sign under reversal of orientation, reflecting the fact that the signature is
an invariant of oriented manifolds. Consequently the simple-minded average 11)
Pontn (RM z)dL O.
On the other hand, since the Pontrjagin numbers are cobordism invariants, in
case M is compact the absolute value