(^62) 2. KAHLER-RICCI FLOW
An oriented Riemannian surface (M, 9) has a natural complex struc-
ture and 9 is a Kahler metric with respect to this complex structure. In
particular, define the almost complex structure J : TM ---+ TM as counter-
clockwise rotation by 90° with respect to the orientation and metric (clearly
J^2 = -idT M). Since dim IR M = 2, we have dw = 0, which implies \J J = 0.
2. Connection, curvature, and covariant differentiation
In this section we consider the connection, curvature, and covariant dif-
ferentiation on a Kahler manifold (Mn, J, 9). Our emphasis, i:n. the style of
the book by Morrow and Kodaira [275], is to calculate geometric quanti-
ties in local holomorphic coordinates, where the formulas are particularly
elegant and simpler than their Riemannian counterparts. These calculations
shall prove useful in our study of the Kahler-Ricci flow.
The Christoffel symbols of the Levi-Civita connection, defined by( ) \J IC a~"'- 8zf3 a ::::;= -~('Ya L...J f a/J 8z'Y + f^1 a/J 8z'Y a) '
-y=l
etc., are zero unless all the indices are unbarred or all the indices are barred.
To see this, from the Riemannian formula for the Christoffel symbols and
since we are complex linearly extending the covariant derivative, we have
the following using (2.2).LEMMA 2.8 (Kahler Christoffel symbols). Let 9a°i3 be defined by 9-yjJ9a°/3 =
8~. Then, in holomorphic coordinates, we have( 2.3 ) r 'Y a.{3 -- 29 ~ -y8 (~ aza.9f3o -+ ~ 8zf39ao -- azo9a.{3 !! ) -- 9 -y8 aza.9f3o ~ -
and
r'Y a.{3 -- f'Y {3a(in fact, the last equality is equivalent to the Kahler condition).^4 Similarly'Y _ ~ 'Y-s ( a __ a _ a -) _
r a/J - 29 aza.9f3o + 8zf39ao - azo9af3 - 0,r~/J = 0, and so forth.
EXERCISE 2.9. Show thatr^1 a/J -r'Y - af3·
(^4) We leave this as an exercise or see for example Theorem 5.1 in [275].