350 7. THE REDUCED DISTANCE8.2. Bounds for £-Jacobi fields. Let c: > 0 and let 'Ys : [O, f] ----t
M, s E ( -c:, c:) , be a smooth 1-parameter family of £-geodesics. In this
subsection we adopt the notation of subsection 8.1 above. Assume Y (0, s) =
0 for s E (-c:, c:) (for simplicity we may assume 'Ys (0) = 'Y (0) for alls). We
shall estimate from above the norms of £-Jacobi fields Y (r) = Y (r, 0).
By the first variation formula for the £-length and the £-geodesic equa-
tion, we have for s E (-c:, c:),
We differentiate this again to get
(6}£) ('Y) = 2vfr (V'xY, Y) (f) + 2vfr (X, \7yY) (f),
where we used \7yX = 'VxY.
Now the derivative of the norm squared of the £-Jacobi field isd~lr=r IYl2
= d~lr=r IY(r)l~(r) = 2(\7xY,Y) (f) +2Rc(Y,Y) (f)
1
(7.125) = 2 Re (Y, Y) (f) + ...ff ( 6}£) ('Y) - 2 (X, \ly Y) (f),which is expressed in terms of the second variation of £. Let Y be a vector
field along 'Y which satisfies the ODE(7.126)(7.127)(vxY-) (r) =-Re (Y-(r)) + 2 ~Y (r), r E [O,f],
y (f) = y (f).
(The first equation is the same as (7.66).) As in (7.68),(7.128) ly (r)l
2
= ~ IY (f)l^2.In particular, Y (0) = 0 = Y (0).
Now we further assume that the 'Ys are minimal £-geodesics for each
s E (-c:, c:). Let is : [O, f] ----t M be a I-parameter variation of 'Y with
- al -
8
is = Y, is (f) = 'Ys (f) and is (0) = 'Ys (0) ;
S s=Othis is possible because Y (0) = Y (0) and Y (f) = Y (f). Then C (is) ~
C bs) for all s, and equality holds at s = 0. Hence
(6}£) ( 1 ) :S (6ic) ('Y),
where equality holds if Y is an £-Jacobi field. Combining this with (7.125),
we get
d~ lr=r IYl2