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we deduce that
CURVATURE AND COMPLEX SINGULARITIES 437
ez(h) (Pz(t, h))
--cte / halt311haz32tz1"" hazk-Bzk-ltzghazgBztzg
C te Roo’" Rz_ ltz-ao
by (1.5) and a straightforward skew-symmetry argument
C te It(RM)
by (1.9). Aside from the explicit determination of constants (a non-trivial matter!),
this implies Weyl’s formula (1.10).
We remark that (1.10) may be extended to manifolds M C Su in spheres
(Weyl, loc. cir.). We also note that the fact that Pz(h)
-- .
contains hz, only in the
quantities hzu h,z,u may be deduced from the observation that Pz(h) is invariant
under substitutions hz, hzgu for (g) an arbitrary proper orthogonal
matrix.
(b) Tubes and the Gauss-Bonnet theorem. We should like to make a few
observations concerning (1.10). To begin with, as already remarked by Weyl,
the first step (1.12) expressing the volume vol rr(M) in terms of the second
fundamental form of M in R N is "elementary calculus." The deeper and more
interesting aspect, which we only outlined, is that the functions P(h) are expressible
in terms of the R’s, and are therefore intrinsic invariants of the
Riemannian metric. The simplest special case of this is that of an arc C in the
(x, y)-plane: it asserts the area of a strip of width r about C depends only on the
length of C and not on its curvature. This invariance under bending may be
illustrated by considering the figures
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