Page 84 - Curvature and Homology

66 XI. TOPOLOGY OF DIFFERENTI ABLE MANIFOLDS
the path of integration. Instead- of considering the z-plane we may
consider a surface S on which the function w(z) is defined and single-
valued. The surface S is called the Riemann surface of the algebraic
function w(z). It can be shown that the Riemann surface of any algebraic
function is homeomorphic to a sphere with g handles. On the other
hand, we may consider such a surface and ask for those functions on the
surface which correspond to single-valued analytic functions in the
x-plane. In this way, we obtain a classification of analytic functions
according to their Riemann surfaces. Moreover, the behavior of the
integrals of the algebraic functions may be determined from a knowledge
of the functions themselves, as well as the topology of the surface. This
is Riemann's approach to the study of algebraic functions and their
integrals. Since the first betti number of a compact Riemann surface
S is 2g, it can be shown that the periods of an everywhere analytic
(henceforth, called holomorphic) integral on S are linear combinations
of 2g periods. By constructing integrals with prescribed periods on 2g
independent 1-cycles of a compact Riemann surface S, it can be shown
that the de Rham cohomology group D1(S) is isomorphic to the group
H1(S). This is de Rham's isomorphism theorem for compact Riemann
surfaces.
Consider now the linear differential form

over a Riemann surface S and define the operator * by

That *a has an invariant meaning over S is easily seen by choosing a
conformally related coordinate system (x', y'):

that is
ay
-- _- -- _-- ay
ax
ax
ax, ayt ' ay, ax' '
and checking the transformation law. The operator * has the following
properties:
(i) *(a+P) =*a+*& *(fa) =f(*a),
(ii) **a = *(*a) = - a,

(iv) a A *a = 0, if and only if, a = 0.