1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

296 7. THE REDUCED DISTANCE

By the mean value theorem for integrals, there exists T* E (0, f) such that

(iii) Let 'fl : [o, 2-JTJ --+ M be a minimal geodesic from p to q with

respect to the metric g (f). Then

D

3. The first variation of £-length and existence of £-geodesics

Now that we have defined the £-length, we may mimic basic Riemannian

comparison geometry in the space-time setting for the Ricci flow. We com-

pute the first variation of the £-length and find the equation for the critical

points of£ (the £-geodesic equation). We also compare this equation with

the geodesic equation for the space-time graph (with respect to a natural

space-time connection) and prove two existence theorems for £-geodesics.

3.1. First variation of the £-length. Let (Nn, h (T))' T E (A, n)'

be a solution to the backward Ricci flow. Consider a variation of the C^2 -path

/ : [Ti, Tz] --+ N; that is, let

G: [Ti, Tz] x (-c:, c:)--+ N

be a C^2 -map such that

Gl[T1,T2]x{O} = /.

Convention: We say that a variation G (·, ·) of a C^2 -path I is C^2 if

G ( ~

2

, s) is C^2 in ( O", s).

Define Is~ Gl[ 71 , 72 ]x{s} : [T1, Tz]--+ N for -c; < s < c:. Let

. 8G O/s. 8G O/s
x (T, s) =;= OT (T, s) = OT (T) and y (T, s) =;=as (T, s) = as (T)

be the tangent vector field and variation vector field along / 8 ( T) , respec-
tively. The first variation formula for £ is given by