Page 92 - Curvature and Homology

74 11. TOPOLOGY OF DIFFERENTIABLE MANIFOLDS

from which

By (2.8.7), this may also be written as

Two linear operators A and A' are said to be dual if (Aa, p) = (a, A'p)
for every pair of forms a and p for which both sides of the relation are
defined. Thus, the operators d and 8 are dual.
In the same way, we see that, if /3 is of degree p - 1, then

Hence, in order that a be closed, it is necessary and sufient that it be
orthogonal to all co-exact forms of degree p.
The condition is indeed necessary; for, if da = 0, then (a, Sp) = 0
for any (p + 1)-form p. Suppose that a is orthogonal to all co-exact
forms of degree p. Then, (a, Sda) = 0, and so (da, da) = 0. Hence,
from property (i), p. 71, it follows that da = 0.
In order that a form be co-closed, it is necessary and sujickt that it be
orthogonal to all exact forms. It follows that if ar and p are two p-forms,
a being exact and co-exact, then (a, 8) = 0.
We now show that in a compact Riemannian manifold the defznitions
of a harmonic form given by Hodge and Kodaira are equivalent. Assume
that a is a harmonic form in the sense of Kodaira. Then,

Hence, since (da, da) 2 0 and (Sa, Sa) 2 0, it follows that da = 0 and
Sa = 0. The converse is trivial.
In particular, a harmonic function in a compact Riemannian manifold is
necessarily a constant.
We have seen that a harmonic form on a compact manifold is closed.
This statement is false if the manifold is not compact. For, a closed form
of degree 0 is a constant while in En there certainly exist non-constant
harmonic functions.
The differential forms of degree p form a linear space AP(T*) over R.
Denote by AP,(T*), A$(T*) and AMT*) the subspaces of AP(T*)
consisting of those forms which are exact, co-exact and harmonic,
respectively. Evidently, these subspaces are orthogonal in pairs, that is
forms belonging to distinct subspaces are orthogonal. A prform
orthogonal to the three subspaces is necessarily zero (cf. 5 2.10). In other
words, the subspaces A,P(T*), Ai(T*) and AL(T*) form a complete
system in AP(T*). (We have previously written A,P(T*) for AZ(T*)).