Page 127 - Curvature and Homology

3.7. CONFORMAL TRANSFORMATIONS 109
Riemannian manifold M. Then, by Stokes' theorem and formula
(3.5.1), if X is an infinitesimal transformation

Since B(X) is a derivation,

If, therefore, we put
*&x) = - B(x)*,
that is

It follows that the operator &X) is the dual of B(X). One thus obtains

where 5 is the covariant form for X. Since the operators B(X) and d
commute, so do their duals as one may easily see from (3.7.10):

Moreover, if g denotes the metric tensor of M

where the are the coefficients of or in the local coordinates (ui).
The proof of (3.7.11) is a lengthy but entiiely straightforward com-
putation and is therefore left as an exercise for the reader.

Theorem 3.7.1. The harmonic forms on a compact and orientable
Riemannian manifold M are K-invariant diferential forms where K is
the Lie algebra of infinitesimal motions on M[73,35].
The proof depends on the fact that B(X) + B(X), X E K annihilates
differential forms. Indeed, since X is an infinitesimal motion, B(X)g = 0
and, therefore, 65 = 0. Let a be a harmonic form. Then, dB(X)a =
B(X)da = 0 and SB(X)a = - SB(X)a = - B(X)Sa = 0. Hence, B(X)a
is a harmonic form; but B(X)a = di(X)a, from which by the Hodge-
de Rham decomposition of a differential form (cf. 5 2.10), B(X)a = 0.