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492 PHILLIP A. GRIFFITHS
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By computing a standard example we may check that the constant is one. Combining
with (4.40) we conclude that
which establishes (4.28) when n k 1.
The argument for general k is similar using all the equations (4.32) and (4.34)
and will not be given in full detail now.
Appendix to sections 2 and 4. Some general observations on integral geometry.
Upon scanning sections 2 and 4 on integral geometry the reader may suspect
that the various Crofton formulas are different manifestations of the same
basic phenomenon, and we want to explain that now. Given a connected Lie
group G and closed subgroups H and K we denote left cosets by : gH and
x g’K. We assume given an incidence correspondence
I C G/H x G/K
which is invariant under the action of G. In practice I will be the union of Gorbits
but in general will not be acted on transitively by G. 11 Denoting by
and rr2 the respective projections of I onto G/H and G/K the basic operation in
integral geometry is
(4.42) --
(rrl),(vr)
where is a differential form on G/K. If we denote the right side of (4.42) by
I(), then I() takes invariant forms to invariant forms and all of our
integral-geometric formulas arise by evaluating I() over suitable submanifolds
of G/H. Here are some illustrations.
Example 1. Suppose that G E(n) is the real Euclidean group and
G/H IR n is Euclidean space
G/K IR n* is the space of affine hyperplanes.
The incidence correspondence is
I {(x, :): x
Given a curve C in IR" and taking Id:l to be the invariant volume on INn*,
Crofton’s formula (2.1) is just the evaluation of
(4.43) fc I()l"