Page 231 - Curvature and Homology

6.6. COMPLEX PARALLELISABLE MANIFOLDS 2 13

Denoting the common value by h we obtain

Corollary 2. There exzit no holomorphic te71~orfields t of type (,P :) on a
compact Kaehler-Einstein manifold in each of the cases :
(i) q > p and h > 0,
(ii) q < p and h < 0.

In either case, for h = 0, t is a parallel tensor Jield.

6.6. Complex parallelisable manifolds
Let S be a compact Riemann surface (cf. example 1, 5 5.8). The genusg
of S is defined as half the first betti number of S, that is bl(S) = 2g.
By theorem 5.6.2, g is the number of independent abelian differentials
of the first kind on S.
We have seen (cf. 5 5.6) that there are no holomorphic differentials on
the Riemann sphere. On the other hand, there is essentially only one
holomorphic differential on the (complex) torus. On the multi-torus Tn
there exist n abelian differentials of the first kind, there being, of course,
no analogue for the n-sphere, n > 2. The Riemann sphere has positive
curvature and this accounts (from a local point of view) for the distinction
made in terms of holomorphic differentials between it and the torus
whose curvature is zero.
Since the torus is locally flat (its metric being induced by the flat
metric of Cn) the above facts make it clear that it is complex parallelisable.
Indeed, there is no distinction between vectors and covectors in a
manifold whose metric is locally flat. On the other hand, a complex
parallelisable manifold can be given an hermitian metric in ierms of
which it may be locally isometrically imbedded in a flat space provided
the holomorphic vector fields generate an abelian Lie algebra and, in
this case, the manifold is Kaehlerian.

Theorem 6.6.1. Let M be a -complex parallelisable manifold of complex
dimensibn n. Then, by de$nition, there exists n (globally defined) linearly
independent holomorphic vector Jields XI, -.-, Xn on M. If the Lie algebra
they generate is abelian, M is Kaehlerian and the metric canonically defined
by the Xi, i = 1, .-a, n, is locally flat [lo].
Let Or, r = 1, ..-, n denote the 1-forms dual to XI, a*., X,,. Thus,
they form a basis of the space of covectors of bidegree (1,O). In