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CURVATURE AND COMPLEX SINGULARITIES 493
Note the integration of densities, since without absolute values the result would
be zero.
Here is a proof of (2.1) which illustrates the general principles of integral
geometry: By the G-invariance of and the incidence correspondence I, the
integral in (4.43) is a G-invariant density on C. According to the theory of moving
frames 12 this density is expressible in terms of the basic invariants which
describe the position of C in G/H (in this case arclength, curvature, torsion,
.). Moreover, and this is the main point, it is visibly clear that if two curves
C and C; osculate tofirst order at some x0 IR"13, then the densities II()1 on
C and C’ agree at this point. It follows that II()1 is a constant multiple of
arclength ds, which implies (2.1).
In general the order of contact necessary to determine I() at some point
gives a first idea of what sort of formula we will obtain.
Example 2. The Crofton formulas (2.3) and (4.1) may be similarly formulated.
In fact (4.1) is easier, since in the complex case there is no problem with
orientations and we may use the following argument: taking G to be the group
of complex Euclidean motions and
G/K= I,,
G/H "*,
and taking b and 4* to be the respective K/ihler forms on tI;" and "* and
(I/n!) 4*" the volume form, we deduce that
I() (rl).(r*)
is a G-invariant (1, 1) form on (I;n. It follows trivially from Schur’s lemma that
I() is a multiple of 4). For a holomorphic curve C in n, by the Wirtinger
theorem (3.19)
d
vol (C)
is the area of C, and by evaluating a constant we have proved (4.1).
Example 3. This gives the method for establishing the reproductive property
(0.4). Again G is the Euclidean group acting now on IRN, and we first take
to be the triples (x, S, L) where
G/K Ouc(n k, N k, N)
x IRN, S is an (n k)-plane, L is
an (N- k)-plane, and x S c L;
this is a flag manifold. Secondly, we take
G/H (n(n, N)