- £-JACOBI FIELDS AND THE £-EXPONENTIAL MAP 361
To get a feeling of the £-Jacobian, we calculate an example.EXAMPLE 7.103 (£-Jacobian of Ricci fl.at solution). Recall the fact that
if (Mn, 9 ( T) = 9o) is a Ricci flat solution, then an £-geodesic is of the form
r (T) = (3 (2y'T) where (3 (O") is a constant speed geodesic. Then an £-Jacobi
field is of the form
JV (T) = K (2.JT)
where K (O") is a Riemannian Jacobi field along (3 (O") with respect to 90·
H ence, b y c h oosmg. JV n ( ) T =^2 v ;;; Tl df3 ff ((J ( 2.j7) r,;) I ,
d<J 2yT g(O)
.cJv (T) = 2JrJv (2vr),
where Jv is the Jacobian of the Riemannian exponential map of 90· Since
by (7.137),
lim Jv (O") = 1
a---+0+ O"n-1 'we have
lim ,C JV ( T) = 2n
T--+0+ Tn/2Note that for Euclidean space ]Rn we have .CJ v ( T) = 2nTn/^2.
As suggested by Example 7.103 and (7.131), we now prove the following
lemma.
LEMMA 7.104 (£-Jacobian as T ---t 0). Let (Mn, 9 ( T)) be a solution
of the backward Ricci flow with bounded sectional curvature. We have the
following asymptotics for the £-Jacobian at T = 0:
(7.138) lim .CJv (T) = 2n.
T--+0+ Tn/2PROOF. Let Ei ( T) denote the parallel translation of Ef along /V ( T)
with respect to 9 (0). Since JY (0) = 0 and ( V' de;, JY) (0) = (V'vJY) (0) =
Ef, then by the definition of derivative,
. IJY (T) - 2y'TEi (T)Jg(O). IJY (T) - O"Ei (T)lg(O)
hm = hm =0.
T--+0+ 2y'T T--+0+ (}Hence
lim .CJv}T) = lim T-n/^2 det (\2Jr.Ei (T) ,2JrEj (T)) 9 (o))
T--+0+ Tn 2 T--+0+
= 2n.
DFor the proof of the no local collapsing result via the £-distance in
Chapter 8 we need the following properties of the .C-Jacobians.