Page 207 - Curvature and Homology

EXERCISES 1 89

Example 6 in 5 5.1 cannot be given a Kaehler structure except for
P x S1 since in all other cases b, is zero. It may be shown by employing
the algebra of Cayley numbers (cf. V.B.7) that the 6-sphere S8 possesses
an almost complex structure. However, since b,(S6) = 0, S8 does not
have a Kaehlerian structure.
Besides S2, the only sphere which may carry a complex structure is S8.
However, it can be shown that the almost complex structure defined
by the Cayley numbers is not integrable.

EXERCISES

A Holomorphic functions [SO]
1. Let S be an open subset of Cn. In order that f e F (the algebra of difirentiable
functions on S) be a holomorphic function it is necessary and sufficient that

where xi = xi + drlyi. Put f = u + drlo. Then,

au av av au.
= - and - - ay" a = 1,-,n.
axi - - -
axi ayi
These are the Cauchy-Riemann equations. Prove that the holomorphic functions
on S are those functions which may be expanded in a convergent power series
in the neighborhood of every point of S.
If f is a holomorphic function and a = (a1, we., an) E S, the'n, for every
b = (br, ma., bn) E C,, the function

is a holomorphic function in a neighborhood of 2 = 0 E C.
2. (a) Let f be a holomorphic function on the complex manifold M. If, for every
point P with local coordinates (zl, ***, zn) in a neighborhood of Po with the local
coordinates (a1, ***,an), I f(zl, -.., zn) 1 5 I f(al, *-*,an) I, then f(xl, *-,zn) =
f(al, a**, an) for all P in a neighborhood of Po. Hence, if M is compact (and
connected), a holomorphic function is necessarily a constant.
(b) A compact connected submanifold of Cn is a point.