1547671870-The_Ricci_Flow__Chow

2. THE LINEARIZATION OF RICCI

Differentiating at t = 0, we obtain

[X, [Y, Z]] =! <p; [Y, Z] = [ :t ( <p;Y), <p; Z] + [<p;Y, :t (<p; Z)]

= [[X,Y],Z] + [Y,[X,Z]],

which is equivalent to the Jacobi identity.

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To illustrate the basic idea, we first consider the diffeomorphism invari-

ance

R [<p;g] = <p; (R [g])

of the scalar curvature function R. Linearizing this identity, we get

(3.12) DR 9 (£xg) = £xR = \1 xR.

If the variation of g is h, then formula 3. 7 of Lemma 3. 7 may be written as

i. ke

DR 9 (h) = -g Jg (\li\ljhke - \li\lkhje + Rikhje).

Substituting

hij = (£xg)ij = \liXj + \ljXi

and commuting covariant derivatives yields

DRg (£xg) = -2b.. \liXi - 2Rij\liXj + \li\lj\liXj + \li\lj\lj xi

= 2Xi\lj ~j·

Combining this result with equation (3.12), we obtain

2Xi\lj Rij = Xi\liR.

Since X is arbitrary, this proves that the diffeomorphism invariance of R

implies the contracted second Bianchi identity:

. 1
(3.13) VJRij = 2,\liR.

Similarly, recalling formula (A.2) from Section 2 of Appendix A, we

observe that the diffeomorphism invariance (3.11) of the Ricci tensor implies

that

(3.14) [D (Re 9 ) (£xg)]jk = (£x Rc)jk = (X, \1 Re)+ Rik \ljXi + Rji \1 kXi.

On the other hand, Lemma 3.5 implies

[D (Rc 9 ) (h)]jk = ~ (Vi\ljhki + \li\lkhji - b..hjk - Vj\lkhD.

When h = £ x g, this becomes

[D (Re 9 ) (£xg)]jk = Rij \lkXi + Rik \ljXi + ~ (Xi\ljRik +Xi! kRij)

• ~ (\le Rejkixi + \le Rekjixi).