1547671870-The_Ricci_Flow__Chow

5. NOTATION FOR HIGHER DERIVATIVES 285

defined so that whenever (Mn, g) is Ricci-parallel - that is, whenever the
(3, 0) -tensor V' Re vanishes identically, as happens in particular if g is Ein-
stein - the following diagram commutes:

If A is a symmetric (2, 0)-tensor, then the components of the Lichnerowicz

Laplacian 6LA in lo cal coordinates are given by the formula

(A.4)

6LA = ( 6Aij + 2gkp/q~kejApq - leRikAej -leRjkAie) dxi@dxJ.

REMARK A.2. As we saw in Chapter 3, the Lichnerowicz Laplacian is

essentially the linearization of the Ricci flow operator.

REMARK A.3. The Lichnerowicz Laplacian on symmetric 2-tensors is

formally the same as the Hodge-de Rham Laplacian on 2-forms. Indeed, an

easy application of the first Bianchi identity shows that equation (A.3) is

equivalent to

6da = ( 6aij + 2gkp lq Rik£jOpq - le Rikaej - gke Rjkaie) dxi /\ dxj,

which is formally the same as equation (A.4).

5. Notation.for higher derivatives
   A delicate notational issue arises when considering the operators

V'k: coo (T%Mn) -7 coo ( T%+kMn)

defined fork> 1. If A E C^00 (T%Mn) and X1, ... ,Xk are vector fields, we

adopt the convention (which is not universally followed!) that

Y'x1Y'x2···Y'xkA

is the unique (p, q)-tensor field such that

(V' x1 V' x2 · · · V' xkA) (Y1, ... , Yp; th, ... , eq)

= (Y'kA) (X1,X2, ... ,Xk, Y1, ... ,Yp; e1, ... ,eq)

for all vector fields Y1, ... , Yp and all covector fields e1, ... , eq· Note that

( V'k A) (X1, ... 'Xk) ~ V' X1 V' X2 ... V' xkA # V' X1 (V' X2 (-.. (V' xkA) ))

in general. For example, in our convention,

(A.5) Y'x (Y'vA) = Y'xY'vA + Y'vxvA = (Y'^2 A) (X, Y) + (V'A) (Y'xY).