Page 167 - Curvature and Homology

5.1. COMPLEX MANIFOLDS 149

however, are not sufficient to ensure that a separable Hausdorff space
has a complex structure as may be shown by the example of the 4-sphere
due to Hopf and Ehresmann [30]. It is beyond the scope of this book
to display this example as it involves some familiarity with the theory
of characteristic classes.

Examples of complex manifolds :
1) The space of n complex variables Cn: It has one coordinate neigh-
borhood, namely, the space of the variables zl, .-*, zn.
2) An oriented surface S admits a complex structure. For, consider a
Riemannian metric ds2 on S. Locally, the metric is 'conformal', that is,
there exist isothermal parameters u, v such that ds2 = A(du2 + dv2) with
h > 0. We define complex (isothermal) coordinates z, 5 by putting
z = u + iv where the orientation of S is determined by the order (u, v).
In these local coordinates ds2 = h dz d5. In terms of another system
of isothermal coordinates (w, 6), dsZ = p dw d6. Since dw = a dz + b d5
it follows that a6 = a'b = 0, from which, by the given orientation b = 0
and dw = a dz. We conclude that w is a holomorphic function of z.
Hence, condition (ii) for a complex structure is satisfied.
3) The Riemann sphere S2: Consider SZ as C v 00, that is as the one
point compactification of the complex plane. A complex structure is
defined on S2 by means of the atlas:
(U,, 24,) = (C, 6) where i is the identity map of C,

In the overlap I/; n U2 = C - 0, u2ui1 is given by the holomorphic
function 5 = llx.
4) Complex projective space P,: Pn may be considered as the space
of complex lines through the origin of Cn,, (cf. 5 5.9 for details). It is
the proper generalization to n dimensions of the Riemann sphere PI.
5) Let r be a discrete subgroup of maximal rank of the group of
translations of Cn and consider the manifold which is the quotient of
Cn by I'; this is a complex multi-torus-the coordinates of a point of C,
serving as local coordinates of Cn/r (cf. $5.9).
6) Senm1 x S1: Let G denote the group generated by the trans-
/
formation of Cn - 0 given by (a1, ... , zn) + (29, ..- ,229. Evidently,