1547671870-The_Ricci_Flow__Chow

NOTES AND COMMENTARY 315

Parallel translate /3 bt ( 0)) to get a unit vector field U along "It, and
define

Then V is a Jacobi field with V bt (0)) = /3 bt (0)) and V bt (Ct)) = 0.
Because "It is minimal among geodesics from (3 to a ( t), the second variation
formula applied with T ~ ~t yields

0:S1Ct (1VrVl^2 - (R (T, V) V, T)) dT.

Noting that 'VrV = -t;u, we calculate that the first term is

{Ct 2 1 {Ct 1

lo l'VrVI dT =Cf lo dT = Ct.

Since sect (g) > 0, there is E > 0 depending only on (3 c Cs C Mn such that

foCt (R (T, V) V, T) dT 2: fo

1

(R (T, V) V, T) dT

1

1

(Ct - T)

2

>E -- d T>E--Ct -^1.

• o Ct - Ct
  Hence if there is a smooth geodesic loop (3 of length less than 1f / ../K con-
  tained in Cs, we must have
  O < 1 + E - dt


• Ct
  holding for all choices of t E (1, oo ). Since ft ----t oo as t ----t oo, this is
  impossible. D

Notes and commentary

The main reference for comparison geometry is Cheeger and Ebin's book

[27]. A more recent survey is the book [54] edited by Grove and Petersen.