116 2. KAHLER-RICCI FLOW
PROOF. Let ha/j be a real (1, 1)-tensor. Using (2.33), we compute:t (\7 i\17ha/j) = \7 ;s'\J 7 ( :t ha/j) - \7 8 ( ( :t r~<Y.) h pfj) - ( :t r~/3) \7 7hap
= \7 8 \7 7 ( :t ha/j) + \7 8 (gP0-\7 7Raa-h pfj) + gO'P\7 8 Rfjo-\7 7hap
(2.123) = \75\7 7 (:tha/j) + \75\77Raphpfj
- \7 7 Rap\75hpfj + \75Rpfj\77hap·
Taking the complex conjugate of (2.123), we have
(2.124) at a· (\7 7 \75ha"{J) = \7 7 \75 (a at ha/'J ) + \7 7 \75Rpfjhap
- \75Rpfj\7 7 hap + \7 7 Rap\75hpfj·
Next we compute, using (2.123), (2.124) and tracing, that
at a (!::,,ha"{J) = 2at^1 a [ 97 8 (\75\77hafj+\77\75ha"{J) ]
- 2 1978 at a '('7_'7 v 8 v 7 h af3 -+ '7 v 7 '7-h v 8 af3 -)
1
+ 2R81 (\75\77ha/j + \77\75ha"{J)= /::,, ( :t ha/j) + ~R81(\75\77ha/j + \7 7 \75ha"{J).
' 1 ,• '+ 2 (\71\77Raphpfj +.\7 7 Rap\71hpfj + \71Rpfj\7 7 hap)
1 ' '
+ 2 (\7 7 \71Rpfjhap + \71Rpfj \7 7 hap + \7 7 Rap \71h pfj).In particular, simplifying· and taking ha/j to be the Ricci tensor, we have
( :t - /::,,) (!::,,Ra"{J) = /::,, (Rafj78R81) - /::,, (Ra1R7"{J)
1
+ 2R81(\75\77Rafj + \7 7 \75Ra"{J)
1+ 2 (\71 \7 7RapRpfj + \7 7 \71RpfjRap)
- \77Rap\71Rpf'J + \71Rpfj\7 7 Rap·
On the other hand, using (2.103), we compute
a · a
at (Rafj 7 8R;y8) = (\7 7 \75Ra/'J - RafiRpfjry8) R;y5 + Rafj 78 at R18
+ 2Ra"fJ 7 8R8pRp 1..