1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

2. CONNECTION, CURVATURE, AND COVARIANT DIFFERENTIATION 69

Hence

\7 a \7 13a'Y1 ""/'pJ1 ... Jq - \7 f3 \7 aa'Y1 "''YpJ1 ... Jq

= (a13I'~'Yj -^0 aI'~'YJ a'Y1·"'Yj-17/'YH1"·'YpJ1 .. ·Jq·

Now from (2.3) we have at p,

813I'~'Yj - OaI'~'Yj = g^71 >. ( Of30a9'Yj>. - Oa0(39'Yj>.) = 0,

and the lemma follows. D

EXERCISE 2.22. Compute the commutator formula for the operator

[\7 a, V' .s] =€= \7 a V' .s - V' .s\7 a acting on (p, q)-tensors and forms.

SOLUTION TO EXERCISE 2.22. If a is a (p, q)-tensor, then

\7 a \7 .Ba'Yl "''YpJ1 ... Jq - \7 .B \7 aa'Yl "''YpJ1 ... Jq

p

= - L Ra.B'Yiifia'Yl"''Yi-17/'Yi+l"''YpJ1 ... Jq

i=l

q

+ L Ra,871Jja'Yl'"'YpJ1 ... Jj-1ifiJH1 .. ·8q·

j=l

The commutator [ Ll, V' .s] , acting on functions, is

1

Ll\7,Bf = 2 (V'aY'a + Y'aY'a) V'.sf

1 1

= 2V'a\7,BV'af + 2\7.BV'aY'af

1

(2.19) = 2 R'Y.8\7'yf + V.silf.

LEMMA 2.23 (Kahler \7 and Ll commutator). The commutator [\7 a, Ll],

acting on (0, l)-forms, is given by

1 1

(2.20) \7 aLla,B - Ll \7 aa,8 = 2 \7 aR8,BaJ - 2 RaJ \7 8a,8 + Ra"f8.B \7 'Ya5.

PROOF. We compute

2 (\7 aLla,B - Ll \7 aa,B) = \7 a (\7 'Y \7 'Y + \7 'Y \7 'Y) a.s - (\7 'Y \7 'Y + \7 'Y \7 'Y) \7 aa,8

= \7 'Y (\7 a \7 'Y - \7 'Y \7 a) a.B + (\7 a \7 'Y - \7 'Y \7 a) \7 'Ya,8

= \7'Y (Ra"f8.BaJ) - Ra"f'YJ\78a,8 + Ra"f8.8\7'YaJ

= \7 aR8,BaJ - Ra5\7 8a,8 + 2Ra"f8.B\7 'Ya5,

where we used [\7 a, \7 'YJ = 0 and \7 'YRa"f8.B = \7 aR8,B from (2.9). D

REMARK 2.24. By taking the complex conjugate of the lemma above,
we have

(2.21)