1547671870-The_Ricci_Flow__Chow

74 3. SHORT TIME EXISTENCE

Combining this equation with (3.14) and again using the fact that X is
arbitrary, we obtain the following consequence of the second Bianchi identity

(3.15)

Notice that this yields the contracted second Bianchi identity when one
contracts on the indices j, k.
Finally, we observe that the diffeomorphism invariance of the Riemann
curvature tensor implies that

where by formula (A.2), the components of the Lie derivative of the Riemann
tensor are

(3.16a)

(3.16b)

(.Cx Rm)fjk = XP\7 pRfjk + R~jk \liXP +Rf pk \ljXP

+ Rfjp \7 kXP - Rfjk \7 pXc.

On the other hand, Lemma 3.3 implies that

2 [D (Rm 9 ) (h)];jk = \li\ljhk + \li\lkh] - \li\lchjk

• \lj\lihk - \lj\lkhf + \lj\lchik·

Substituting h = .Cxg and commuting derivatives, we find that

2 [D (Rm 9 ) (.Cxg)Jfjk = \li\lj\lcXk - \li\lc\ljXk - \lj\li\lcXk

• \lj\lc\liXk + \li\lk\lcXj - \li\lc\lkXj


• \lj\lk\lcXi + \lj\lc\lkXi - \li\lj\lkXc

• \li\lk\ljXc + \lj\li\lkXc - \lj\lk\liXc


• 2\li\lj\lkXc - 2\lj\li\lkXc.

Thus if we rewrite (3.16) as

Xq\7 q~jkp - \7 iXqRkpj - \7 jXqR~ki }

• \7 kXqRq. ZJp - \lqXpRq ZJ .k


• \7 i ( RkpjXq) - \7 j ( R~kiXq) - \7 kXqR{jp }


• \lqXpR{jk + Xq (\lqRijkp + \liRkpj + \ljR~ki)