1547671870-The_Ricci_Flow__Chow

2. THE LINEARIZATION OF RICCI

defined by

(3.24)

The total symbol of B 9 in the direction ( is the bundle homomorphism

S2T* Mn --+ T* Mn

given by

(a [B 9 ] (()(h))k = lj ( (ihjk - ~(khij).

77

Notice that a [B 9 ] (() is of degree 1 in (. Writing the contracted second
Bianchi identity (3.13) in the form

B 9 (Rc 9 ) = 0

shows that Re 9 belongs to the kernel of B 9. Linearizing, we obtain

(D (B 9 )) [Re (g + h)] + B 9 [((D Re 9 ) (h))] = 0.

Since the differential operator D (B 9 ) is of order 1, its degree 3 symbol is

zero. But it is easy to check that the linear operator B 9 o D (Re 9 ) is of order

3. It follows that the principal (degree 3) symbol

0-[B 9 o D (Re 9 )] (() = 0-[B 9 ] (() o 0-[D (Re 9 )] (()

must be the zero map, hence that

The considerations above have shown that for all (, the following short

sequence constitutes an algebraic chain complex:

(3.25)

0 --7 T* Mn~ S2T* Mn &[D(Rcg)](() S2T* Mn~ T* Mn --7 o.

We shall now show that it is in fact a short exact sequence by proving

that the nontrivial kernel of 0-[D (Re 9 )] ( () (hence the failure of D (Re 9 )

to be elliptic) is due only to the invariance (3.11) of the Ricci tensor under

diffeomorphism. We begin by studying the kernel of a [B 9 ] ((). Given a

nonzero 1-form (, define

and

Kc~ ker (a [B 9 ] (()) <:;;; S2T* Mn.

Clearly, dim Ac = n in each fiber where ( # 0. In any such fiber, the

calculation

1

(a [B 9 ] (() (h), X) = "2 (( 0 X + X 0 ( - ((, X) g, h)