Page 94 - Curvature and Homology

76 11. TOPOLOGY OF DIFFERENTIABLE MANIFOLDS

and therefore, by the completeness of the system of subspaces AP,(T*),
A$(TT) and A&(T*) in AD(T*), a; = ad, a; = a,, a; = a~. We have
proved:
A regular form a of degree p may be uniquely decomposed into the sum

where ad E A$(T*), a, E A$(T*) and a~ E API(T*).
This is the Hodge-de Rham decomposition theorem [39].

2.1 1. Fundamental theorem

At this stage it is appropriate to state the existence theorems of &
Rham [65]-the proofs of which appear in Appendix A.
(i
(R,) Let {q} = 1, ***, b,(M)) be a base for the (rational) p-cych
of a compact dz#eeentiable manifold M and wk (i = 1, ., b,(M)) be bp
arbitrary real constants. Then, there exists a regular, closed p-form a on
M having the wk ar periods, that is

(R,) A closed form having zero perrbdc is an exact form.
We now establish the existence theorem due to Hodge which is at
the very foundation of the subject matter of curvature and homology.
There exists a unique harmonic form a of degrree p having arbitrarily
assigned periods on bp independent p-cycles of a compact and orientable
Riemannian manifold.
Indeed, let a be a closed p-form having the given periods. The
existence of a is assured by the first of de Rham's theorems. By the
decomposition theorem a = ad + am (Since or is closed, a, is zero and
consequently a is orthogonal to A$(T*)). Since ord E A$(Tt) its periods
are zero. Hence the periods of a, are those of a. The uniqueness
follows from (R,) since a harmonic form whose periods vanish is the
zero form.
Let M be a compact and orientable Riemannian manifold. Then, the
number of linearly independent real harmonic forms of degrec p is equal to the
pth betti number of M.
For, let vd denote the harmonic p-form whose periods are zero except