Page 235 - Curvature and Homology

In terms of the metric g, the connection defined in § 5.3 is given by the
coefficients

Differentiating with respect to 5' we conclude that Rf,,. = 0.

Theorem 6.7.3. A complex parallelisable manifold has a natural hermitian
metric of zero curvature.
Since

(Dj denoting covariant differentiation with respect to the given con-
nection), it follows that

Multiplying these equations by I;, and taking account of the relations

we conclude that
Dj 5':) = 0, Y = 1, -*, n.

Thus, we have

Corollary. A complex parallelisable manifold has a natural hermitian
metric with respect to which the given field of frames is parallel.
The results of this section are interpreted in V1.G.

6.8. Compact complex parallelisable manifolds
Let M be a compact complex parallelisable manifold. Since the
curvature of M (defined by the connection (6.7.1)) vanishes, the con-
nection is holomorphic; hence, so is the torsion, that is, in the notation
of 5 5.3
d"D =0, i= 1;-,n