Page 218 - Curvature and Homology

200 VI. CURVATURE AND HOMOLOGY: KAEHLER MANIFOLDS

Hence,
Theorem 6.1 .l. A Kaehler manifold of constant curvature is locally flat
provided n > 1.
If, instead of insisting that all sectional curvaturb at a given point
are equal, we require that only those determined by any two orthogonal
vectors in the tangent space at each point are equal, the same conclusion
prevails, since the bundle of orthogonal frames suffices to determine the
Riemannian geometry. For complex manifolds, however, it is natural
to consider only those Zdimensional subspaces of the tangent space
defined by a vector and its image by the linear endomorphism ] giving
the complex structure. Indeed, to each tangent vector Xp at a point P
of the hermitian manifold M, one may associate the tangent vector ( JX)p
at P orthogonal to Xp. The section determined by these vectors will be
called a holomorphic section since it is defined by the complex structure.
We shall denote the sectional curvature defined by the holomorphic
section determined by the vector Xp by R(P, X) and call it the
holomorphic sectional curvature defined by X,.
We seek a formula in local complex coordinates for R(P, X). To
begin with, if
a a a
x = P = his + ti* xi,
then, from (5.2.4)

Hence, from (1.10.4)

where qi = 6 1 5' and qp = - GI e*. Now, it is easy to see that

Consequently,
Rij*kl* p [j* fk 5.'
R(P,X) =
gij* gkl* P ti* Sk Sz*
which, by reasons of symmetry, may be expressed in the form