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446 PHILLIP A. GRIFFITHS
I (2.2) n(LCqC)dL= I F*(o)2/ (-O12),
and we shall evaluate the right hand side by integration over the fibre. For this
it is convenient to parametrize C by arc length; i.e., to give C by a vectorvalued
function x(s) where x’(s)l] 1. The Fr6n6t frame is defined by
dx * *
(.O le
de* * *
0.) 12e2
where o* ds and O*lZ (s)ds with (s) being the curvature. (z) The fibre
7r-l(x(s)) consists of lines whose frames are {x; e, e} where
as pictured by
x x(s)
el= cos0e* + sin0e
e2 -sin 0 e*l + cos 0 e*z
Using (0, s) as coordinates on B (and dropping reference to F*), we have
O01 (dx, el) cos 0 ds,
019.
(de1, ez)
(-sin 0 e*l + cos 0 e, ez)dO mod ds
dO mod ds.
Thus dL cos 0 dO/ ds, and the right hand side of (2.2) is (a)
Ic (I0 ,cos OldO)ds=2 le ds
2/(C),