Page 203 - Curvature and Homology

5.9. EXAMPLES OF KAEHLER MANIFOLDS 185

ds2 = p2dz d2 where p is a real, positive function (of class 00) of the
local coordinates x, y(z = x + iy), i = GI. The fundamental 2-form
52 = (272) p2dz A dE is the element of area of S. Clearly, dl2 = 0 since
dim S = 2. The real unit tangent vectors which are given by

determine a sub-bundle 8 of the tangent bundle called the circle bundle.
We define a differential form o of bidegree (1,O) by the formula
u = e-fpp dz.
Evidently, (e(tp), w) = 1. Conversely, o is uniquely determined by
the conditions: (i) it is of bidegree (1,O) and (ii) its inner product with
the vectors of 8 is 1. Consider the 1-form 8 on 8 defined as follows:

One may easily check that 8 is real and satisfies the differential equation

In fact, 8 is the only real-valued linear differential form satisfying this
differential equation with the property that 8 E - dtp (mod (dz, dE)).
Hence, 8 is globally defined in 8, independent of the choice of local
coordinates. Moreover,
de = - a2 log p d2 A d5.
2i-
az a2
Now, the Gaussian curvature K of S is given by

4 a2 log p
K= ---
~2 az a5
from which
de = KQ.
It is known that a compact Riemann surface can be given an hermitian
metric of constant curvature and that such surfaces may be classified
according to whether K is positive, negative or zero.
Incidentally, besides the Riemann sphere (K > 0) and the torus
(K = 0) any other compact Riemann surface can be considered as the
quotient space of the unit disc by some Fuchsian group.