Page 86 - Curvature and Homology

68 11. TOPOLOGY OF D~FFERENTIABLE MANIFOLDS

df is we (of bidegree (1,O) cf. 5 5.2). A linear differential form a on S is
said to be a holomorphic dtperenticzl if, in each coordinate neighborhood U
it is the differential of a holomorphic function in U. A linear differential
form a is locally exact, if and only if, da = 0. Locally, then a = df and in
order that f be holomorphic *df = - idf or *a = - iu. A differential
form satisfying this latter condition is said to be pure. Hence, a linear
dzperential form of class 1 is holomorphic on S, if and only if it is closed
and pure (cf. 5 5.4). We remark that if a is holomorphic, it is a harmonic
form. This is clear from the previous statement.
The formal change of variables z = x + iy, 5 = x - iy and the
resulting equations *dz = - idz, *d5 = id5 clarify the nature of
pureness: a is pure, if and only if, it is expressible in terms of dz only.
A differential form of class 1 will be called a regular dz~erential form.
Now, the regular harmonic forms on a compact Riemann surface S
form a group H(S) under addition. It can ,be shown that if u is a closed
linear differential form on S, then there is a unique harmonic 1-form
homologous to a, that is H(S) is isomorphic to the de Rham cohomology
group D1(S). This is Hodge's theorem for a compact Riemmn surface.
The proof is based on a decomposition of a into a sum of two forms,
one of which is exact and the other harmonic. (More generally, a l-form
on a Riemannian manifold may be decomposed into a sum of an exact
form, a form which may be expressed as *df for some f and a harmonic
form (cf. 5 2.7). This is the decomposition theorem applied to 1-forms).
The de Rham isomorphism theorem together with the Hodge theorem
for compact Riemann surfaces implies that the first betti number of a
compact Riemunn surface is equal to the number of linearly independent
harmoni'c I-forms on the surface.

2.7. The star isomorphism

The geometry of a Riemann surface is conformal geometry. As a
possible generalization of the results of the previous section, one might
consider more general surfaces, for example, the closed surfaces of
5 1 .l, the geometry being Riemannian geometry. One might even go
further and consider as a -replacement for the Riemann surface an
n-dimensional Riemannian manifold. To begin with, consider the
Euclidean space En and let (ul, ***, un) be rectangular cartesian coor-
dinates of a point. Let f be a function defined in En which is a potential
function in some region of the space. In the language of vector analysis,
div grad f = 0, (2.7.1)