1547671870-The_Ricci_Flow__Chow

2. THE LINEARIZATION OF RICCI 75

and compare terms, we get

0 = [D (Rmg) (.Cxg)J;jk - (.Cx Rm)fjk

-\Ji [ ( Rjpk - Rkpj - Rjkp) xq]

-2Xq (\7 qRijkp + \liRkpjq + 'VjRpkiq)
Now let p E Mn be arbitrary. Since X is an arbitrary vector field, we
may first prescribe X (p) = 0 and \liXj (p) = 9ij (p), obtaining

0 = - (Rjeki - Rkeji - Rjkei) + (Rmj - Rkeij - Rikej).

Recalling the symmetries Rijke = - Rjike = - Rij£k = Rkeij, we see that this

implies the first Bianchi identity

(3.17) 0 = ~jke + ~kej + ~ejk·

Then if we choose X to be an element of a local orthonormal frame field and
use the first Bianchi identity to cancel terms, we obtain the second Bianchi
identity

(3.18)

2.3. The principal symbol of the differential operator Re (g).

We shall now explore the principal symbol 0-[D (Reg)] of the Ricci tensor,
regarded as a nonlinear partial differential operator on the metric g.
Recall that the formal adjoint of the divergence
(3.19a)
(3.19b)

bg : C^00 (S2T* Mn)----+ C^00 (T* Mn)

(bgh)k ~ -gij\lihjk

with respect to the L^2 inner product

(3.20) (V, W) = { (V, W) dμg

}Mn

is the linear differential operator

(3.21a) s; : C^00 (T* Mn) ----+ C^00 (S2T* Mn)

(3.21b) (s;x)jk ~ ~ ('VjXk + \lkXj) = ~ (.Cxtt9)jk.

In other words, s;x is just a scalar multiple of the Lie derivative .Cxtt9 of

g with respect to the vector field x" which is g-dual to X. The total symbol

of s; in the direction ( is the bundle homomorphism

(J [s;J ((): T* Mn----+ S2T* Mn

that acts by

(3.22)