Page 219 - Curvature and Homology

Suppose that R(P, X) is independent of the tangent vector X chosen
to define it. Then, the curvature tensor at P has the representation

where k = k(P) denotes the common value of R(P, X) for all tangent
vectors X at P. For, by assumption, the equation

is satisfied by the 2n independent variables (e, p*). Hence, since
both sides of (6.1.1) are symmetric in the pairs (i, k) and (j, I) we have
the desired conclusion.
Theorem 6.1.2. If the holomorphic sectional curvatures at each point of a
Kaehler manifold are independent of the holomorphic sections passing
through the point, they are constant met. the manifold.
We wish to show that the function k appearing in (6.1.1) is a constant.
By assumption, the curvature tensor has this form at each point of M.
Transvecting (6.1.1 ) with gk'* we derive

that is M is a '(Kaehler-) Einstein' space (cf. 5 6.4). Hence, from
(5.3.29) and (5.3.38) the 1"' Chern class of M is given by
+-"+la.
4rr
Since ab is closed.

from which by corollary 5.7.2, dk must vanish for n 2 2..
If at each point of a Kaehler manifold the holomorphic sectional.
curvature is independent of the tangent vector defining it, the manifold
is said to have constant holomorphic curvature.
Theorem 6.1.3. P, may be given a metric g in terms of which it is a
manifold of constant holomorphic curvature.
Indeed, we give to P, the Fubini metric g of example 5, 5 5.9:

in the coordinate neighborhood U,.