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436 PHILLIP A. GRIFFITHS I0 is A is not a rearrangement of B | _+ is the sign of permutation taking ai to/3i otherwise, and where C te C(n, l) is a suitable constant. Setting ,(M) It(R,)dM the formula is (1.10) vol rr(M) C(l, n)txt(M)rm + . /=0 0(2) We shall sketch Weyl’s proof, deferring the detailed argument until section 3c in which the complex case will be discussed. Points in ’r(M) are y x+ t.e.. It follows from the structure equations (1.3) and (1.4) that dy (toe- tto.)e + (dt. + tto..)e. with repeated indices always being summed. Letting dV denote the volume element in IR N and dt dtn / 1/ / dtN, we infer that where (1 11) Pt(t, k) By Fubini’s theorem dV= / ( (-h.t.)toz) / ’ Pl(t, h)dM dt (- 1)t I! A,B h11"1"" htmt"l"" tm. To evaluate the inner integral we utilize Weyl’s notation {f} for the spherical average 465) f(t)dt, and also his explicit evaluation (Weyl, loc. cit., page (1.13)
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 l i’/ /////// /i///l
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For k we have (4.33) CURVATURE AND
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As a consequence of (5.15), CURVATU
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