1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

68 2. KAHLER-RICCI FLOW

PROOF. Using the fact that the Christoffel symbols are zero unless all

of the indices are unbarred or all are barred, we compute

and

\7 a \7 °(Ja'Y = aaa°(Ja'Y - r~'Ya°f3a8'

\7 $ \7 aa'Y = a$ ( 8aa'Y - r~'Ya8).

Hence, using (2.4), we have

\7 a \7 °(Ja'Y - \7 $ \7 aa'Y = a$r~'Ya8 = -R!°f3'Ya8,

which is (2.16). Equation (2.17) is the complex conjugate of (2.16).

and

If a is a (1, 1)-tensor, then we compute

\7 °!3a'Y8 = a°f3a'Y8 - I'$8a'Ye'

\7 aa'Y8 = 8aa'Y8 - r~'Ya'IJ8

\7 a V' °!3a'Y8 = aa ( a°f3a'Y5 - I'$8a'Ye) - r~'Y ( a°f3a'IJ8 - I'$8a'IJe) '

\7 $ V' aa'Y8 = a$ ( 8aa'Y8 - r~'Ya'IJ8) - r$8 ( 8aa'Ye -:-r~'Ya'IJe).

Hence, after some cancellations, we obtain in holomorphic coordinates { z°'}

at a point p where it is unitary,

\7 a \7 °!3a'Y8 - \7 $ \7 aa'Y8 = -8aI'$8a'Ye + a$r~'Ya'IJ8

= Ra$c:8a'Ye - Ra°f3'Yfia'IJ8.

D

LEMMA 2.21 (\7 a and \7 f3 commute). We have [\7°''\713] = 0 acting on

tensors of any type. That is, if a is a (p, q)-tensor, then

PROOF. 'We have

\713a 'Yl "''Yp81 -···8q - -- 813a 'Yl "''Yp81 -.. ·8q - - r'IJ /3'Yj a 'Yl "''Yj-l'IJ'YHl "''Yp81 -.. ·8q - ·

It is easiest to compute the second covariant derivative in normal holomor-
phic coordinates centered at any given point p EM, where r~/3 (p) = 0.
In these coordinates we have at p,

\7 a \713a 'Yl"''Yp81 -.. ·8q - -- {) a (013a 'Yl"''Yp81 -.. ·8q - - r'IJ /3'Yj a 'Yl"''Yj-l'IJ'Yj+l"''Yp81···8q - -)

= a a 813a 'Yl "''Yp81 -.. ·8q - - (a a r'IJ f3'Yj ) a 'Yl '"'Yj-l 'IJ'Yj+l "''Yp81 -.. ·8q -·