Page 129 - Curvature and Homology

3.7. CONFORMAL TRANSFORMATIONS 111

The integral of the left side of (3.7.15) over M vanishes by Stokes'
theorem. Hence, integrating (3.7.15) gives

Thus,

and so, since a and fl are arbitrary
e(x), + #(x)~ = (I - :) 86 + a.

Let M be a Riemannian manifold, Co(M) the largest connected group
of conformal transformations of M and Io(M) the largest connected
group of isometries of M. (Note that L and K are the Lie algebras of
Co(M) and Io(M), respectively.) We shall prove the following:

Theorem 3.7.4. Let M be a compact Riemnnian ntanifold. If Co(M) #
Zo(M), then, there is nu hamtonic form of degree p, 0 < p < n (n = dim M)
whose length is a non-zero constant [78].
Since a harmonic form on a compact Riemannian manifold is invariant
by Io(M), a harmonic form on a compact homogeneous Riemannian
manifold (cf. VI. E) is of constant length. (A Riemannian homogeneous
mansjCold is a Riemannian manifold whose group of isometries is transi-
tive.) Hence, as an immediate consequence of theorem 3.7.4 we have

Theorem 3.7.5. Let M be a compact homogeneous Riemannian mat$old.
If Co(M) Sf. Io(M), then M is a homology sphere [78].
Since we are interested in connected, groups, the hypothesis of
theorem 3.7.4 may be replaced by the fdlowing: Let M be a compact
Riemannian manzjold admitting an injinitesinal non-isometric conformal
transformation. We may also assume that M is orientable; for, if M is
not orientable, we need only take an orientable two-fold covering space
of M.
Proof of Theorem 3.7.4. Let or be a harmonic form of degree p. We
shall first prove
(&x) a, 0(X) a) = 0.
Since or is closed, 6(X) or = di(X) a. On the other hand, since a is
co-closed, M(X) o: = &X) 601 = 0. Thus,