Page 85 - Curvature and Homology

dy,
Since df = ( af / ax) dx + (af/ 9) we can define the operator *d for
functions by

Define
(*d)a = - d(*a)
for 1 -forms.
If we put

then

A function f of class 2 is called harmonic on S if Af vanishes on S.
Locally, then

A linear differential form a of class 1 on S is called a harmonic form if,
for each point P of S there is a coordinate neighborhood U of P such
that ar is the total differential of a harmonic function f in U. This implies
that *a is closed. In fact, a = df and d*df = 0 in U, that is d*a = 0.
Conversely, da = 0 implies that a = df, locally (cf. 4 A. 6). Moreover,
d*a = 0 implies that d(*df) = 0. Hence, f is harmonic. We have shown
that a linear differential form a of class 1 is harmonic, if and only if,
da = 0 and d*a = 0.
A harmonic differential form a = p dx + q dy on S, that is a form
which satisfies da = 0 and d*a = 0 defines a holomorphic function
p - iq (locally) of a = x + iy (i = 1/ - 1). Indeed,

aP 34
0 = d*a = (= + 5) A dy,
dx
and so we have locally (splay) = (aqlax) and (aplax) = - (aqlay),
which are the Cauchy-Riemann equations for the functions P and - q.
(A function f of class 1 is holomorphic on S if locally f(x, y) = u(x, y) +
iv(x, y) and the functions u and v satisfy the Cauchy-Riemann
equations). It is an easy matter to show that f is holomorphic on S,
if and only if *df = - idf, that is, if and only if, the differential