1547671870-The_Ricci_Flow__Chow

2. THE LINEARIZATION OF RICCI 71

Since [U, V] = 0, we have

d d rb 1/2

dt Lt( "It) = dt la g (U, U) du

= ~lb (U U)-1/ 2 ag (U U) d

(^2) a g ' at ' u

• 1b g (U, U)-^112 g (U, \luV) du
  i rb ag rb

= 2.la at (S,S) ds+ la (S,\lsV) ds

= ~ 1b h(S,S) ds + ( (S, V)I~ -1b (\lsS, V) ds).

D

2. The linearization of the Ricci tensor and its principal symbol

2.1. The symbol of a nonlinear differential operator. Let V and

W be vector bundles over Mn, and let

L : C^00 (V) ---> C^00 (W)

be a linear differential operator of order k, written as

L (V) = L Laaav,

lal:Sk

where La E horn (V, W) is a bundle homomorphism (namely, a linear map
when restricted to each fiber) for each multi-index ex. If ( E C^00 (T* Mn),
then the total symbol of L in the direction ( is the bundle homomorphism

o-[L](() ~ L La(IIj(aJ).

laJ:Sk

Note that o-[L] (() is a linear map of degree at most kin(. The principal

symbol of L in the direction ( is the bundle homomorphism

0-[L] (() ~ L La (IIj(aJ).

lal=k

The principal symbol of a differential operator L captures algebraically those

analytic properties of L that depend only on its highest derivatives. The

following basic property of the total symbol will be useful: if X is another

vector bundle over Mn and

M: C^00 (W) ---> C^00 (X)

is a linear differential operator of order £, then the symbol of M o L in the

direction ( is the bundle homomorphism

(3.9) o-[Mo L] (() = o-[M] (() o o-[L] (() : V---> X