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448 PHILLIP A. GRIFFITHS
where el/k e. is the plane spanned by el and e.. We have encountered this map
in section ld above, and from the discussion there it follows that the invariant
me.asure on G(2, 3) is
dH to13 /k
If C is given parametrically
_
by the unit vector e(t), then we attach to C the
Fr6n6t frame {el, e, e} where
e*l e,
de] tO*l.e z,*" i.e., to] 0
de$ --60 * 12e * + oo* 23 3.
Then o. +-[[e’(t)[[dt is the element of arc length, and o*ia *(t)o*i where
* is essentially the curvature. As before we have a diagram
B --) G(2, 3)
C
where B C C G(2, 3) is the incidence manifold {(e, H) e C H), and the
left hand side of (2.3) is a constant times I. [0913 / 0923]. TO iterate the integral
we parametrize all great circles passing through e(t) by frames {e, e,
where
el e
e =cos0e + sin0e
ea sin0e+cos0e.
As before
1(,013 / (.0231 [Ix’(t)ll jcos O] dO/X dr,
and (2.3) follows, where the constant r is determined by taking C to be a great
circle.
As an application of (2.3) we give what is the first relation between curvature
and singularities. Recall that if f(zl, z) 0 defines a complex analytic curve V
passing through the origin in , then setting S {]Zll / ]z2l } the intersection
V V f3 S defines a closed curve in the 3-sphere whose knot type
reflects the topological structure of the isolated singularity which V has at the
origin6). We shall be concerned with the total curvature of a closed curve C in
IR and shall prove the following results: ) The total curvature satisfies
(2.4) I1 ds 2
with equality if and only if C is a convex plane curve (FencheO; and if C is
knotted then