1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

8 1. RICCI SOLITONS

Now we proceed to systematically differentiate the function f up to fourth

order, commute pairs of covariant derivatives, trace, and apply (1.24) and

the second Bianchi identity to relate derivatives of the curvature and f.

We begin with considering three derivatives of f since for one derivative

there is nothing to do, and a pair of covariant derivatives acting on a function

commute. ·Since \Jk [\Ji, 'Vj] f = 0, the only nontrivial commutator is

\Ji\Jj\Jkf - \Jj\Ji\Jkf =[\Ji, 'Vj] \Jkf.

By (1.24), the commutator formula, and \Jg= 0, we have

(1.26)

Taking the trace by multiplying by gjk and using the contracted second

Bianchi identity, we get

(1.27)

Note that (1.26) is antisymmetric in i and j, and in particular, (1.27) is the

only equation obtained by tracing (1.26).

Next we consider equations obtained by commuting four derivatives off.

The only essentially new equation is obtained by considering \7 i [\7 j, \7 k] \7.e.f.

The quantity \7 i \7 j [\7 k, \7.e.] f is zero and [\7 i, \7 j] \7 k \7 d yields only a stan-

dard commutator formula. We have

which implies

and hence

(1.28)

First we trace by gij. · Commuting derivatives and applying the contracted

second Bianchi identity yield

1 E

b..Rk.e. - 2 \7 k \7.e.R + 2~k.e.mRim - RkmRm.e. + 2 Rk.e.

= \7 iRikR.m \7 mf

= ( -\7.e.Rkm + \7 mRk.e.) \7 mf ·

Since (1.28) is antisymmetric in j and k, we have that tracing (1.28) by gjk
is zero, whereas tracing by gik is equivalent to tracing by gij. The remaining
trace is by gie. We leave it as an exercise for the reader to show that taking
this trace yields (-\JjRkm + \JkRjm) \7 mf = 0, which is nothing new since
it follows directly from (1.26).