Page 105 - Curvature and Homology

Denote by Q the operator on 1-forms defined by

and assume that the quadratic form

is positive definite. Since the second term in the integrand of (3.2.5) is
non-negative we conclude that

from which a = 0. Since a is an arbitrary harmonic 1-form we
have proved

Theorem 3.2.1. The first betti number of a compact and orientable
Riemannian manifold of positive definite Ricci curvature is zero [4, 621.
If we assume only that (Qa, a) is non-negative, then from (3.2.5)
(Qa, a) as well as DjaiDjai must vanish. It follows that Djai vanishes,
that is the tangent vectors

are parallel along any parametrized curve ui = uC(t), i = 1, a*., n.
A vector field with this property is called a parallel vector field.
Theorem 3.2.2. In a compact and orientable Riemannian manfold a
harmonic vector field for which the quadratic form (3.2.6) is positive semi-
definite is necessarily a parallel vector field [q.

Theorem 3.2.3. In a coordinate n&ghborhood of a compact and mientable
Riemannian manvold with the local coordinates ul, ..a, un, a necessary and
suflcient condition that the 1-form a = aiduf be a harmonic form is given by

Piai - gfk Dk D, ai = 0 (3.2.7)
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Clearly, if a is harmonic, (3.2.7) holds. Conversely, if the 1-form a
is a solution of equation (3.2.7) then, by (3.2.3), Aa A +a = 0. Hence,

0 = (Aa, a) = (da, da) + (Sa, Sa),
from which da = 0 and 6a = 0.