1547671870-The_Ricci_Flow__Chow

178 6. THREE-MANIFOLDS OF POSITIVE RICCI C URVATURE

PROOF. By applying the second Bianchi identity and commuting covari-
ant derivatives, we compute

and then apply the second Bianchi identity again to obtain

9 pq...., v p Re jqk -- 9 pq 9 em(-......, v k R· Jqmp - ......, v m R· Jqpk ) -- ......, v k j Re - ....,eR· Y 3k·

This lets us rewrite the formula for 6.Rfjk given above as

6.Rfjk = -\7i\7kR] + \7i\7eRjk + 'V/'hRf-\7/veRik

• gpq { R;ijR;qk + R;iqR]rk + R;ikR]qr - R~irRJqk }.

-R;jiR;qk - R;jqRfrk - R;jkRfqr + R~jrRiqk

Now the first Bianchi identity shows that

which implies that 6.Rfjk may be written as

(6.12a)

(6.12b)

(6.12c)

6.Rfjk = - \7i\7kR] + \7i\7eRjk + 'Vj'VkRf

• 'Vj'Ve~k - RiR]rk + RjRfrk

• gpq { -RijpR;qk + R;ikR]qr + R~jrRiqk }.

• R~irRJqk - R;jkRiqr

On the other hand, rewriting the commutator term gf-P (\7 j \7 i - \7 i \7 j) Rkp

in formula (6.2) yields

(6.13a) :tRijk = - \7i\7kR] + \7i\7eRjk + 'Vj'VkRf-'Vj'Ve~k

(6.13b)

The reaction-diffusion equation for the Riemann curvature tensor now fol-

lows from comparing terms in formulas (6.12) and (6.13). D

We define the (4, 0)-Riemann curvature tensor by Rijke ~ 9emR'':Jk, so

that R1221 > 0 on the round sphere. The following observation then follows

immediately from the lemma above.