1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

352 7. THE REDUCED DISTANCE

REMARK 7.89. Note that for Euclidean space, £-Jacobi fields satisfy

IY(r)l^2 = const·r. In particular, d~loglY(r)l^2 =~(and .e is constant

along £-geodesics).

PROOF. Since g (r) has nonnegative curvature operator, Hamilton's ma-

trix inequality holds and we have for any r E [O, r],

H ( X, Y) ( r) + ( ~ + T ~ 7 ) Re ( Y, Y) ( r) ~ 0.

Since Re ~ 0 and ly (r) 1

2

= ¥ IY (r)l^2 , from r ::::; (1 - c) T, we get for

r E [O, r],

Then (7.129) implies

dd I _ IYl^2 ::::; ( \:o: f

7

\/'TR(!' (r), r) dr + 1) IY (!')1

2

"T T=T Cy "T lo "T

::::; (~.e (!' (r) 'r) + 1) IY ~)12'

since I is a minimal £-geodesic. Hence

-d d I _log IYl^2 ::::; -:::^1 (2 -.e (!' (r) , r) + 1. )

"T T=T "T C

Finally, we leave it as an exercise to check that when T = oo, one can

in essence take c = 1 in the inequality above. D

8.3. The £-exponential map. The £-exponential map

£exp: TM x [O,T)---+ M

is defined by

£exp (V, r) ~ £ expv (r) ~ 1v (r) ,

where IV (T) is the £-geodesic with lim 7 -+o y'T~;_ (r) = V E TpM (and
IV (0) = p). Given r, define the £-exponential map at timer

£7exp: TM---+ M

by

£7 exp (V) ~IV (r).