Page 104 - Curvature and Homology

86 III. RIEMANNIAN MANIFOLDS: CURVATURE, HOMOLOGY

For,
JM 8a A *l = (8% 1)
= (a, dl)
= (a, 0) = 0.

In the sequel, we employ the notation (t, t') to mean the (local)
scalar product of the tensors t and t' of type (0, s) in case t and t' are
simultaneously symmetric or skew-symmetric, that is

If t and t' are skew-symmetric tensors, (t, t') = (a, a') where a and
a' denote the corresponding s-forms (cf. II.A.2). From (2.7.11)

(a, d) = JM (a, d) *l.
Now, consider the integral

whose value is zero by (3.2.4). Indeed, if we put fl = d(a4g),

Then,

where we have put
Dj = gjk Dk.
Hence,
D)
- jM aigjk Dk Dj a, *1 = jM~j d *I,
mi
and so if a is a harmonic 1-form (3.2.2) becomes