Page 254 - Curvature and Homology
P. 254
Hence,
and if I is the identity operator on forms
a log
glm*(D1 + pal ' I) +akl...k,m*i;...ia = 01 Pal = - '
Thus, for a harmonic form of bidegree (p, q) with coefficients in B
Consider the l-form
5' = Sm*dgm
of bidegree (0,l) where
It is easily checked that it is a globally defined form on M. We compute
its divergence:
- 65' gh*D1f,* = G(+) + X (6.14.3)
where
1 (6.14.4)
G(+) = glrn*[(Da + Pal *I Dm*#akl...kpi;...iG 1 $akl...kpi;...~
and
1
*a
= -glm*Dm*dakl...kpi~...i~ . ~,*+~k~...k~ii...i;
Formula (6.14.3) should be compared with (6.5.3).
Note that equations (6.14.1)-(6.14.4) are vacuous unless q 2_ 1.
Now, by the Hodge-de Rham decomposition of a 1-form
