486 B. OTHER ASPECTS OF RICCI FLOW AND RELATED FLOWSThus, considering Rep as a 2-tensor on P and changing its covariant
derivative to the one with respect to gp in the formula (B.20), we have\l/vjH = \Ji\Jkhkj - Di (Rep )jv
- (rp )0 (Rep )vv - (rp )~v (Rep) jk
= \Ji\Jkhkj - Di (Rcp)jv
+ hij (Rep )vv - hf (Rep )jk.Commuting derivatives and applying the Codazzi and Gauss equations, we
compute\Ji'VjH = \Jk\Jihkj - (RcM)iehej - (RmM)ikjehke -Di (Rcp)jv
+ hij (Rep )vv - hf (Rep )jk
= \J k \J khij - \J k (Rm P )ikjv- (Hhu - hte) hej - (hiehkj - hijhke) hke
- (Rcp)iehej + (Rmp)vievhej - (Rmp)ikjehke ·
- Di (Rcp)jv + hij (Rcp)vv - hf (Rcp)jk
= b:.hij - Hhtj + lhl^2 hij - Dk (Rmp )ikjv
- hki (Rmp )vkjv + hkk (Rmp )ivjv - hk (Rmp )ikj£
- (Rep) ie hej + (Rm p) viev hej - (Rm p) ikj£ hke
- Di (Rcp)jv + hij (Rcp)vv - hf (Rcp)jk,
where htj ~ hikhkj and in the third line (Rm p )ikjv dxi ® dxk ® dxj is con-
sidered as a 3-tensor. This is known as Simons' identity. Hence, under the
mean curvature flow8 k£
ot hij =\Ji 'VjH - Hhjkg hei + H (Rmp )vijv= b:.hij - 2Hhtj + lhl^2 hij
- Dj (Rcp)iv + Dv (Rcp)ij -Di (Rcp)jv
- (Rep )ie hej + (Rm p )vi£v hej - (Rm p )ikj£ hke
hij (Rcp)vv - hf (Rcp)jk
hki (Rmp )vkjv - hk (Rmp )ikj£,
where we used the second Bianchi identity:
D