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CURVATURE AND COMPLEX SINGULARITIES 451
the function f,(s) has for sufficiently small e 0 two critical values 0, S corresponding
to the two roots of
(
s
0 f(s) s Z(So)-- er(So) + O(s).
We note that s is approximately equal to 2e(so)/r(so). Since
and
f(s) sT"(So)- ere(So) + O(s)
f’(o) -,(So) o,
f’(2e,(So)/r(So)) e,(So) 0
we may assume that both these critical values are non-dengenerate. In summary,
we have shown that in a neighborhood where (s)r(s) 4: O, slight perturbations
of the osculating 2-planes give height functions having two local nondegenerate
critical values in addition to the absolute maximum and minimum.
Thus n(H 71 y(C)) -> 4 on an open set and so equality cannot hold in (2.4).
Now we have proved that C is a plane curve, and a similar argument shows
that either r’ (s) -= 0, in which case C is a circle, or r’ (s) 0 in which case C is a
general convex plane curve. Q.E.D.
(c) The kinematic formula. We shall discuss briefly the reproductive property
(0.4) of the curvature integrals in Weyl’s tube formula (0.10). For this it is
useful to pretend for a moment that we don’t know Weyl’s formula but do know
the Gauss-Bonnet theorem (1.17), and pose the following question: Suppose
that Mn C IR N is a compact manifold and L t(N k, N) varies over the
Grassmannian of affine (N k)-planes in IRN. Outside a set of measure zero in
(m(N k, N) the intersection M fq L will be a smooth manifold of dimension
n k in L IR ,ta) and we may ask for the average
x(M L)dL
of the Euler characteristics of these intersections.
By the Gauss-Bonnet theorem this question is equivalent to determining the
average
(2.6) I (f , I’ (Rt )) dL
where I, (Rt r ) is the Gauss-Bonnet integrand for M L. According to two
of our general principles, the expression for (2.6) should first of all be local--i.e.,
should be valid for a small piece of manifold as well as a global compact
one--and secondly should be expressed as an integral over M of a polynomial
of degree n k in the entries of the curvature matrix Rt of M. Recalling our
discussions at the beginning of l(b) and the end of 2(a), we may give an
heuristic argument for the existence of a formula of the type