- CONNECTION, CURVATURE, AND COVARIANT DIFFERENTIATION 65
which is equivalent to the Ricci form being closed: dp = 0. We may express
(2.6) as
p = -vCf8fJlogdet (g'YJ).
The complex scalar curvature is defined to beR...,.... ...!... g a'iJR a(3· -
The complex scalar curvature is one-half of the Riemannian scalar curvature.
(Exercise: Prove this.)NOTATION 2.15. In this chapter, R always denotes the complex scalar
curvature whereas in the rest of the book, R always denotes the Riemannian
scalar curvature. In this chapter we shall usually refer to the complex scalar
curvature simply as the scalar curvature.Holomorphic coordinates {za} are said to be unitary at a point p if
g ( 8 ~CL , o~f3) (p) = 5a(3. That is, { 8 ~CL (p)} :=l is a unitary frame for Tl,O Mp.
In such a frame we have at p,
n n
(2.7) Rafi= LRa'i]oJ and R= LRaa·
8=1 a=l
Note that for the standard Euclidean metric
1 n
gc = 2 L (dwa ® dvP + diiP ® dwa)
a=lon ccn the new coordinates {z(Y, ~ 0w(Y,} :=1 are unitary.
Given z E T^1 ,^0 M - {o}' let x ~ Re(Z) =! (z + .Z) E TM. The
holomorphic sectional curvature in the direction Z is defined to be(^2 · 8 ) K c (Z) =. Rm(X,JX,JX,X) 1x14
Rm(z+z,A(Z-Z) ,A(Z-Z) ,z+z)
4IZl4
_ Rmc (Z,Z,Z,Z)
1z14Hence Kc (Z) = Rmc (V, V, V, V), where V ~ Z/ IZI. In particular, if
{za} is unitary at p, then Kc ( 8 ~CL) = Raaaa· We say that the holomorphic
sectional curvature is positive (respectively, nonnegative) if Kc (Z) > 0
(respectively, Kc (Z) 2:: 0) for all Z E T^1 ,o M - { o}. By (2.8), positive
(respectively, nonnegative) Riemannian sectional curvature implies positive
(respectively, nonnegative) holomorphic sectional curvature.
Given Z, WE T^1 ,oM - {o}, let X ~ Re(Z) and U ~ Re(W). The
(holomorphic) bisectional curvature in the directions ( Z, W) is defined