- INCOMPRESSIBILITY OF BOUNDARY TORI
torus with
area AFIGURE 33.3. The submanifolds CA,B, TA,p, SA,p, and SA,p·247Let rv denote the restriction of r to V and let Drv denote its gradient. Since
IDrvl ::; l'Vrl = 1, we have by the co-area formula (see Proposition 27.30) that for
a.e. p E [0, PA,B]
(33.67)
A(p)=j dμ?: j IDrvldμ= 1
15
L(AA, 15 n{r=p})dp= 115
L(p)dp
AA,p AA,p 0 0since AA,15 n {r = p} = SA,p·
Given a.e. p E [O,pA,B] and a loop SA, 15 C SA, 15 as above, there exists an
embedded loop SA, 15 (0) ETA such that
(expv) -1 (SA,- -15 ) = {pv(x): x E SA,15(0)}.
Consider the cylinder (topologically an embedded annulus)
(33.68) AA, 15 ~ { expv ({rv(x): x E SA,15(0), r E [O,p]})}.This set is the union of the normal geodesics to TA of length p ending in S A, 15.
Let
sA, 15 (p) ~ AA, 15 n TA,p, p E [O, p],
which is a loop. Define
(33.69) L 15 (p) ~ Lengthg(t)(SA, 15 (p))