340 7. THE REDUCED DISTANCE
COROLLARY 7. 72. The variation of the £-length on a steady gradient
Ricci soliton Re (go)+ \J\J Jo = 0 is given byr2V¥ I da .)
8£ (!)(Y) = - Jo \ 2\J ~~ da + V Jo, Y da,where \J is the Levi-Civita connection for g (0). Hence the £-geodesic equa-
tion is
da 1
\J do:-d do- (} + - 2 VJo = 0.This also implies that the following directional derivative vanishes:- da (ldal- +Jo =0.
2
)
da da g(O) ·That is, along an £-geodesic, the square of its speed with respect to a plus
the potential function is constant.
EXERCISE 7.73. Show that the above equation is equivalent to (7.35):
2.
0 = Di Z - ~ \JR + a Re ( Z) ,where Z ~ .JT1J;. Note that on a gradient soliton ~\JR= Re (V !).
In the rest of this subsection we consider £-geodesics on the cigar on
R^2. The scalar curvature of the cigar solution to the backward Ricci fl.ow
~~ = 2Rc,
. dx^2 + dy^2
g (x, y, T) =;=_ e-4T + x2 + y2'
is given by
4
R (x, y, T) = 1. +eT 4 ( 2 X +y 2)"Let r^2 = x^2 + y^2. The £-length of a radial path I ( T) , 0 :S T :S f, with
r(T) ~r(/(T)), is
C (?) = [ .fi ( 1+ :."r' + c•}+ r' (~~ )')aT
Define s (T) ~ sinh-^1 (e^27 r (T)) (which is the distance to the origin with
respect to g ( T)) so that
C('y) = [ (4sech^2 s+ (~; -2tanhs)') ./TdT.