1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

298 7. THE REDUCED DISTANCE

3.2. The £-geodesic equation. The £-first variation formula leads

us to the following.

DEFINITION 7.18 (£-geodesic). If"! is a critical point of the £-length

functional among all C^2 -paths with fixed endpoints, then "! is called an

£-geodesic.

By the £-first variation formula,

COROLLARY 7.19 (£-geodesic equation). Let (Nn, h (T)), T E (A, D),

be a solution to the backward Ricci flow. A C^2 -path "( : h, T2] --+ N is an

£-geodesic if and only if it satisfies the £-geodesic equation:

1 1

(7.32) \lxX-2\1R+2Rc(X) + 2Tx = 0,

where X (T) ~ ~; (T).

For the four terms in (7.32), (1) is the usual term in the geodesic equa-

tion, (2) comes from the variation of R in .C, (3) comes from g 7 h, and (4)

comes from y'7 in .C via integration by parts.

In local coordinates, the £-geodesic equation is

d2"(i i d"(j d"(k 1 i. i. d"(k 1 d"(i

(7.33) 0= dT2 +rjk("!(T),T) dT dT -2hJ\ljR+2hJRjk dT +2T dT'

where 'Yi =xi o "!·

We find it convenient to use the notation £ for the covariant derivative

along the curve /· Multiplying (7.32) by T yields

D T

(7.34) Vr dT ( y!iX) - "2 \1R+2yfiRc ( yfiX) = 0.

Since the covariant derivative along the curve can be written as

D

dT V = \lxV,

we may write

D

Vr dT ( vfiX) = vfi\1 x ( y!iX) = \1 ftX ( yfiX)

along "( ( T) , where the last two terms require extending V ( T) = y'7 X to a

vector field in a neighbor hood of "! ( T). Note that

\1 ftX (vfiX)"= T\lxX + Vr (d~ Vr) X

1

= T\lxX +

2

x,

which is different from y'T\1 v-rxX because y'7 must be differentiated along

the curve. Using the convention (7.18) and Z (o-) ~ d~~) = y'TX, we get

0"2

(7.35) \lzZ -

8

\lR+ o-Rc(Z) = 0.