1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

342 K. IMPLICIT FUNCTION THEOREM

Combining this with (K.32), we have

:s ls=O (6.g,hfs)'y = (6.g,h V)

7

• 6.g (!^13 ) r (h);Q V

0

ij 8 ( o 'Y o 'Y ) or o Jf3

+ g v oy8 r (h)C</3 - oyC< r (h)/38 oxi oxJ

+ g ijv8( -r(h)-r QC: r(h)c /3 )or off3

8 -0 x'. -0 xJ ..

We may further rewrite this and simplify as

(K.33)

:s ls=O (6.g,hfs)'Y = (6.g,h V)^7 - ( 6. 9 (Jf3) +I' (h)~ 8 gij :;)I' (h):/3 V

0

ij 8 ( 0 ( 'Y 0 'Y ) 0 f

0

0 Jf3

+ g v oy8 r h)C</3 - oyC< r (h)/38 oxi oxJ

• gijV

8

(r (h)Jc: r (h)~ 0 - r (h):c: r (h)~ 8 ) ~:: ~~;

= (6. 9 ,h V)^7 - (6. 9 ,hf)/3 r (h):/3 V^0

+ v8 (RN)-Y gijor of

13

.

(^80) /3 ox' oxJ
Note that the terms forming the expression on the RHS of (K.33) are not the com-
ponents of a tensor.
Using (K.33) we compute for V independent of s that
(K.34) :S ls=O (V, 6. g,hfs) J;h
Since
and
= :S ls=O ((ho-y O fs) (V

0

O fs) (6.g,hfs)'Y)

= ho-y V

0

:s ls=O (6.g,hfs)'Y + V

0

( 0 ~ 0 h 0 -y) V

0

(6. 9 ,hf)'Y

• ho-y ( V^13 o~/3 V

0

) (6.g,hf)^7

_ h vo (A v)'Y h vc:v8 (RN)-Y ij or of

13

• C<"f Ll.g,h + q fo/3 g oxi oxJ
  
  
  • ho-y V^0 (6.g,hf)/3 I' (h)J13 V^8 + V^0 (
    0
    ~ 0 h 0 13) V^0 (6.g,hf)/3
  
  
  
  

+ ho-y ( V^13 o~/3 V

0

) (6.g,hf)^7.

i oy/3 v^0 + r (h)^0 f3o v^0 = (vf*hv)/3^0 '