1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

8. £-JACOBI FIELDS AND THE £-EXPONENTIAL MAP 353

EXAMPLE 7.90 (£-exponential map on a Ricci fl.at solution). To get

a feel for the £-exponential map, we first consider a Ricci fiat solution

(Mn,g (T) =go). Here, by (7.23), for VE TpM,

,C exp (V, f) =exp ( 2J:fv) ,

where exp is the usual exponential map of (M, g 0 ) with basepoint p. Note
that for a Ricci fl.at solution, the £-exponential map has the scaling property:

£exp (V, f) =£exp ( cV,;)

for any c > 0. However this is not true for general solutions of the Ricci fl.ow.

The £-exponential map at f = 0 is related to expg(O) which is the usual

(Riemannian) exponential map with respect to the metric g ( 0).

LEMMA 7.91 (£-exponential map as f---+ 0). Let (Mn, g (T)), TE [O, T],

be a complete solution to the backward Ricci flow with bounded sectional

curvature. Given V E TpM, as f ---+ 0, the £-exponential map tends to the

Riemannian exponential map of g (0) in the following sense:

(7.131) !im £exp (

1
'= V, r) = expg(o) (V).
T---+0 2yT
Prom the proof we can see that the convergence in (7.131) can be made into
C^00 -convergence.

PROOF. Motivated by the Ricci fl.at case, we define the path f3 : [O, 1] ---+

Mby

f3 (p) ~£exp ( ~ V, p^2 f) = "(_1 v (p^2 f),

2yT 2../'F

so that f3 (1) =£exp c~ V, f). (Note that f3 depends on f but we do not

emphasize this in our notation.) We have

df3 d ( ) d"( ...-1.,, v

(7.132) -d (p) = -d "(_1 v (p^2 f) = 2pf ~-.fi (p^2 f).

p p 2../'F T

Hence the £-geodesic equation. (7.32) becomes

O = \7 1 d,e (-l_df3) -~\7R + 2Rc (~ df3) + ~~ df3.

2pr dp 2pf dp 2 2pT dp 2p^2 T 2pT dp

Multiplying this by 4p^2 f^2 yields for p E [O, 1],

(7.133) \7 d,e-df3 -2p 2-2 T \7R+ 4 -pTRc (df3) -d = 0.

dp dp p

The covariant derivative and Ricci tensor are with respect tog (p^2 f). Since

d"f_1_v 1

T---+0 lim VT ~-.fi T (T) = 2y EV, T