1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

302 7. THE REDUCED DISTANCE

equation:

d2 ';JO ~ (-o ) d';ji d';jj d2 ((12) 0 (- ) (Cl) 2

dCl2 + ~ rij^0 (3 dCl dCl = dCl2 4 + I'oo /3 (Cl) 2

o::;;i,j::;;n

1 1 ((1)2

= 2 - 2 (er:) 2 = O.

(This last equation justifies defining the time component of '/3 (Cl) as Cl^2 / 4,
and in particular, the change of variables Cl = 2y'T.) For the space compo-
nents, the geodesic equation with respect to f' says that for k = 1, ... , n,

d2 '/Jk -k d'/Ji d'/Jj

(^0) = dCl2 + L rij dCl dCl
o::;;i,j::;;n
d2 {Jk k d{Ji d{Jj -k d{Ji d'/JO -k d'/JO d'/JO
= dCl 2 + L rij dCl dCl +^2 L rw dCl dCl + r^00 dCl dCl ·
l::;;i,j::;;n l::;;i::;;n
This is equivalent to
0 = (~)
2
d
2
"/ (r(Cl)) + "" r~. (~ d/i (r(Cl))) (~ d/j (r(Cl)))
2 dr^2 ~ iJ 2 dr 2 dr
1::;;i,j::;;n

• ~ (~~ (r(Cl))) +2 L Rf (~~~i (r(Cl))) (~)-~ (~)2\7kR,
  I::;;i::;;n
  which, after dividing by r = Cl^2 / 4, implies
  d21k d1i d1j 1 ( d1k )
  o = dr2 (r(Cl)) + L rrj dr (r(Cl)) dr (r(Cl)) + 2r dr (r(Cl))
  l::;;i,j::;;n


• 2 ~ ~ Ri kdli( - r(Cl)) - -\7^1 k R.
  1::;;i::;;n dr 2
  That is, in invariant notation and with X ~ ~, we have
  1 1

\7xX--

2

\7R+2Rc(X) +-X = 0,

2r

which is the same as (7.32). Thus £-geodesics correspond to geodesics de-
fined with respect to the space-time connection. In particular, I (r) is an
£-geodesic if and only if (3 (Cl) ~ I ( Cl^2 / 4) is a geodesic with respect to the

space-time connection '\7. Since f'~b = limN-+oo Nf'~b' we also conclude that

the Riemannian geodesic equation for the metric hon Nn x (0, T) (defined

in Exercise 7.4) limits to the Cl = 2y'T reparametrization of the £-geodesic
equation as N ----+ oo.

EXERCISE 7.23 (Motivation for change of time variable). Show that if

'/3: [O, <7] ----+ N x [O, T] is a geodesic, with respect to the connection '\7, with