1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

328 7. THE REDUCED DISTANCE

may write formula (7.65) at smooth points off as

(7.96)

( V'V'f +Re- 2 ~g) (Y, Y) (f)

=-

2

~fo

7

y'TH(X,Y)dr+1

7

~IV'xY+Rc(Y)-; 7 Yl

2

dr.

EXERCISE 7.53. What does the trace of this equality say?

We have the following property of the reduced distance f.

LEMMA 7.54 (Reduced distance as r ---+ 0). Given V E TpM, let f1V :

[O, T] ---+ M be the £-geodesic with limT--70+ vr~; ( r) = v. Then

(7.97) T--70+ lim f ( /1V (f) , f) = IVl~(o) ·

PROOF. Since limT--70 vrdJ: ( r) = v and '"YVl[o,7] is a minimal £-geodesic

when f is small (see Lemma 7.29), we have

lim f('"Yv(r),r)

T-t0+

= lim \;: {7 Vf (R('"Yv(i),i)+ lddf1v (7)1

2

) di

T-t0+ 2y T lo T g('Y(i'),1')

= hm..^1 r,; 1T v r:::: T. -;::;^1 I v^12 g(O) dr -= I v^12 g(O) •

T-tO+ 2yr 0 r

D

We conclude this subsection with a few exercises concerning the reduced

distance f,.

EXERCISE 7.55 (£-triangle inequality). Show that for any path f1

[ri, r2]---+ M,

L (11 (r2), r2) - L (11 (r1), r1)

= 2y72f (11 (r2), r2) - 2y7if (11 (r1), r1)

1

72 I d

1

2

= £ (11) -

71

VT V'f - d~ dr.

REMARK 7.56. Given points (ih, f1) and (112, f2), we may define

J:d ( ( ii_i, f1) , ( i]2,T2)) = L(q 1 ,7 1 ) (c°fa, f2)

~inf{£ (11): f1 (fi) = if.i, i = 1, 2}.

Then the above formula implies

J:d ((p, 0), (q1, r1)) + J:d ((q1, r1), (q2, r2)) 2: J:d ((p, 0), (q2, r2)).

Equality holds if and only if for some minimal £-geodesic f1 : [ri, r 2 ] ---+ M

we have ~; = V'f along ')1, where f is defined with respect to the basepoint

(p, 0).