Page 128 - Curvature and Homology

110 111. RIEMANNIAN MANIFOLDS: CURVATURE, HOMOLOGY

Corollary. In a compact and orientable Riemannian manifold the inner
product of a harmonic vector Jield and a Killing vector Jield is a constant.
In fact, if a is a harmonic l-form and X an element of K, 0 = B(X)a =
di(X)a.
The corollary may be generalized as follows:

Theorem 3.7.2. The inner product of a K-invariant closed I-form and
an element X of K is a constant equal to < X, H[a] >.
For, 0 = B(X)a = di(X)a. By the Hodge-de Rham decomposition
of a 1-form, a = df + H[a] for some function f, from which 0 = B(X)a
= B(X)df = di(X)df. Hence, (X, df) = k = const. We conclude that
(5, dn = J*k = 0 since (5, dn = (65,n = 0.
Let X be an element of the Lie algebra L of infinitesimal conformal
transformations of M. Then, equation (3.7.11) reduces to

in view of formula (3.7.4), and we have the following generalization of
theorem 3.7.1 :

Theorem 3.7.3. Let M be a compact and orientable Riemannian manvold
of dimension n. Then, a harmonic k-form a is L-invariant, if and only if,
n = 2k or, a is co-closed [35].

Corollary. On a compact and orientable 2-dimensional Riemannian
manifold the inner product of a harmonic vector field and an infinitesimal
transformation defining a I-parameter group of conformal trattsfmmations
is a constant.
This is clearly the case if M is a Riemann surface (cf. Chap. V).
Since formula (3.7.12) is required in the proof of theorem 3.7.5
and again in Chapter VII a proof of it is given below:
Applying B(X) to (a, 8) = g41j1- giJp %l..,d BUl..Jpl we obtain

We also have
O(X)*l = - Sf *I.

From (3.7.13) and (3.7.14), we obtain