1547671870-The_Ricci_Flow__Chow

1. THE EVOLUTION OF CURVATURE 179

COROLLARY 6.14. Under the Ricci fiow, the (4, 0)-Riemann curvature

tensor satisfies the following reaction-diffusion equation:

(6.14a) :t ~jke = 6.~jke + gPq (RrjpR,.qke - 2R;ikRjqre + 2RpireRjqk)

(6.14b) - (Rf Rpjke + RjRipke + R~Rijpe + R~~jkp).

There is a nice way to rewrite the evolution equation (6.14) in terms of
a (4, 0)-tensor B defined by

(6.15) Bij. k£ ..!... -=--gPTgqs ~Lip]q D .. Rk r s e -- -Rq pij RP q£k'

(The minus sign here, which does not appear in Hamilton's original paper

[58], occurs because we are using the opposite sign convention for ~jke·)

Note that B is quadratic in the Riemann curvature tensor and satisfies the

following algebraic identity

(6.16)

LEMMA 6.15. Under the Riccifiow, the (4,0)-Riemann curvature tensor

evolves by

(6.l 7a)

(6.l 7b)

a

ot ~jke = 6.~jke + 2 (Bijke - Bij£k + Bikje - Biejk)

• (Rf Rpjke + RjRipke + R~Rijpe + R~Rijkp).

PROOF. First observe that the last two terms on the right-hand side of

(6.14a) may be directly expressed in terms of B:

• 2gpq R;ikRjqr£ = -2gpq ~pkrRjq£r = 2Bikje,

2gpq RpireRjqk = 2gpq RiperRjqkr = -2Bi£jk.

This leaves us with gPqRrjpRrqk£· Applying the first Bianchi identity gives

gPq RrjpR,.qke = gPq grs R,.pjiRsqke

= gPqgrs ( -R,.jip - R,.ipj) ( - Rsk£q - Rseqk)

= -Bjike + Bjilk + Bijk£ - Bijek,

whence identity (6.16) implies that

gPq RrjpR,.qke = 2 (Bijke - Bijek).

0

REMARK 6. 16. Observe that although B does not in general satisfy the

first Bianchi identity, the tensor C does, where

Cijke ~ Bijke - Bijek + Bikje - Biejk·

Similarly, although Bjike #--Bijke in general, one has Cjike = -Cijke·