Page 130 - Curvature and Homology

112 111. RIEMANNIAN MANIFOLDS: CURVATURE, HOMOLOGY
Applying (3.7.12) and (3.7.16) we obtain

:)
= (1 - jM8f (a, e(x) a) $1.

From now on, we assume that a is not only harmonic but is also of
constant length, that is, (a, a) is constant. Hence, B(X) (a, a) = 0,
and so, from (3.7.13)
<o(x) a, a) = - P-- 6~ (a, a). (3.7.18)
n
Substituting (3.7.18) into (3.7.17) we obtain

If 2p 5 n, the right hand side of (3.7.19) is non-positive; but the left
hand side is non-negative. Consequently, B(X) a = 0 and by (3.7.18)
either 86 = 0 or a = 0. If X is not an infinitesimal isometry, 86 # 0.
We have therefore proved that if M admits an infinitesimal non-isometric
conformal transformation, then there is no harmonic form of constant
length and degree p, 0 < p 5 42. If a is a harmonic form of constant
length and degree p > n/2, then its adjoint *a is a harmonic form of
constant length and of degree n - p < 42. This completes the proof.
By employing theorem 3.7.5, it can be shown that M is, in fact,
isometric with a sphere (cf. 111. F).

3.8. Conformal transformations (contin ukd)

In this section we characterize the infinitesimal conformal trans-
formations and motions of a compact and orientable Riemannian
manifold M as solutions of a system of differential equations on M.
Moreover, we investigate the existence of (global) 1-parameter groups
of conformal transformations of M and find that when the Ricci curvature
tensor is positive definite no such groups except {e} exist.
For a l-form a on M we define the symmetric tensor field