View PDF - Project Euclid

CURVATURE AND COMPLEX SINGULARITIES 447
completing the proof of Crofton’s formula (2.1).
For future reference we note that the central geometric construction in
the argument is the incidence correspondence I c IR 2 ((1, 2) defined by
{(x, L) x L}. There are two projections
I
IR (1, 2)
and B rSI(C). The basic integral in Crofton’s formula is
c (),(dL),
and as will emerge later the main ingredient which enables us to systematically
evaluate such integrals is invariance under a suitably large group (cf. sections
4a and 4b). )
Finally we remark that on general grounds we may easily deduce that
n(L t C)dL is first of all additive in C, and then by passing to the limit that it
is an integral
Ic
f(x, K(s),
of some function of the curvature and its derivativesindeed, any Euclidean
invariant is of this general type. The main geometric point is that the average
n(L C)dL is a bending invariant, and therefore does not involve the curva-
ture and its derivatives.
(b) Application of Crofton’s formula to total curvature. Of course there are
Crofton formulas existing in great generality5); here we should like to observe
that (2.1) remains valid in the elliptic non-Euclidean case. Explicitly, let C be a
curve lying in the unit 2-sphere S and denote by G(2, 3) the great circles on S
parametrized by the planes H through the origin IRa. Then we claim that the
relation
(2.3) r J n(H 0 C)dH l(C)
is valid.
For the proof we consider the fibration
o%0(IR) Grt(2, 3)
given by
{el, ez, ea} el / ez