Page 93 - Curvature and Homology

2.10. DECOMPOSITION THEOR,EM 75

2.10. Decomposition theorem for compact Riemannian manifolds

Let be a p-form on a compact, orientable Riemannian manifold M.
If there is a p-form a such that Aa = p, then, for a harmonic form y,

Therefore, in order that there exist a form a (of class 2) with the
property that Aa = p, it is necessary that p be orthogonal to the sub-
space A~(T*). This condition is also sufficient, the proof being given
in Appendix C. The original proof given by Hodge in [39] depends
largely on the Fredholm theory of integral equations.
The dimension of A&(T*) being finite (cf. Appendix C) we can find
an orthonormal basis ..., 9h} for the harmonic forms of degree p:

Any other harmonic p-form may then be expressed as a linear com-
bination of these basis forms. Let a be any p-form. The form

is harmonic and a - a~ is orthogonal to A&(T*). In fact,

It follows that there exists a form y such that Ay = a - a,. If we set
ad = day and a, = ady, we obtain ad + a, = a - a,, that is

where ad E A $(T*), ad E /I\g(T*) and a~ E A s(T*). That this decom-
position is unique may be seen as follows: Let a = a; + a: + 4
be
where a; E A;(T*), ad E AC(T*) and a& E A~T*) another decom-
position of a.