Page 245 - Curvature and Homology

some C, with the flat metric. It is not difficult to see that the Ricci
curvature of g is given by
R(z,a) = - ---- <O. (6.1 1.14)
(1 - zq2
From (6.1 1.13) and (6.11.14) we obtain immediately that the scalar
curvature is - 2. Thus, g has constant negative curvature, that is g is
a hyperbolic metric.
Another example is afforded by the higher dimensional analogue,
namely, the interior of the unit ball Zr-, I xi l2 < 1 with the hyperbolic
metric
E I dzi l2 - C zi la ZI I dzj la + I ZIZi dzi I2
I
I
(1 - C xi 12)=

6.12. Euler characteristic
In the previous section we considered manifolds M on which N 2 n
holomorphic functions f'(r = 1, -.., N) are 'locally' defined. Mare
precisely, in a coordinate neighborhood U of M we assumed the
existence of N independent holomorphic 1-forms ar satisfying d'ar = 0.
Now, in this section, we assume that on the complex manifold M there
exists N 2 n 'globally' defined holomorphic differentials

aT = a':' dz', r = 1, -, N, rank (a(:)) = n everywhere,
which are simultaneously d'-closed. The fundamental form

of M is then closed and of maximal rank. The distinction made here is
that we now have a globally defined Kaehler metric

In terms of the curvature of this metric, and by means of the generalized
Gauss-Bonnet theorem, if M-is compact

where x(M) denotes the Euler characteristic of M. Moreover, x(M)
vanishes, if and only if the nth Chern class vanishes. Incidentally, the
vanishing of x(M) is a necessary and sufficient condition fcr the existence
of a continuous vector field with no zeros (on M).