1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

346 7. THE REDUCED DISTANCE

where L ('-y) denotes the length of I· Integrating by parts on (7.119), we may
express this as

I (V, W) = - 1b (V -y V -y V + R (V, 'Y) 'Y, W) ds.

Recall that a vector field J along/ is a Jacobi field if

\l-y\l-yJ +R(J,"f)'Y = 0.

(If J vanishes at the endpoints of/, then I (J, W) = 0 for all W) Equiva-

lently, a Jacobi field is the variation vector field of a 1-parameter family of

geodesics. Given a unit speed geodesic/: [a, b] -+ M, the set of Jacobi fields

along/ is isomorphic to T'Y(so)M x T'Y(so)M, for any so E [a, b]; each Jacobi

field J is determined by the initial data J (so) = Jo and (V-yJ) (so) = Ji for

Jo, Ji E T'Y(so)M·

The Index Lemma says that if/: [a, b] -+Mis a unit speed geodesic

without conjugate points, then among all vector fields along / perpendic-

ular to "f with prescribed values at the endpoints, the unique such Jacobi

field minimizes the index form; i.e., given A E T'Y(a)M and B E T'Y(b)M

perpendicular to "f, the Jacobi field J with J (a)= A and J (b) = B satisfies

(7.120) I (J, J) :::; I (W, W)

for all W perpendicular to 'Y and such that W (a) = A and W (b) = B.

Equality in (7.120) holds if and only if W = J. In particular, for any W =/=- 0

perpendicular to 'Y with W (a) = 0 and W (b) = 0, we have I (W, W) > 0.

If / has no conjugate points in the interior, but possibly one at b, then

(7.120) still holds although J may not be unique. Furthermore, we have

I (W, W) ;:::: 0 for any W perpendicular to 'Y with W (a) = 0 and W (b) = 0,

where equality holds if and only if Wis a Jacobi field.

8.1. £-Jacobi fields. Let / : [O, f] -+ M be an £-geodesic, where
f E (O,T), and let X (T) ~~~be its tangent vector field.
DEFINITION 7.82 (£-Jacobi field). An £-Jacobi field along an £-geodesic
/ is the variation vector field of a smooth 1-parameter family of £-geodesics

/s, s E (-c:, c:), for some c: > 0, all defined on the same time interval as

/O =/.

Let X (T, s) ~ ~~, Y (T, s) ~ ~~s, and let Y (T) ~ Y (T, 0) be an £-Jacobi

field along f. Using the £-geodesic equation (7.32), we compute

\Ix (\lxY) =\Ix (\lyX) = R (X, Y) X + \ly (\lxX)

= R(X, Y) x + \ly (!vR-2Rc(X) - _!__x).

2 2T