Page 95 - Curvature and Homology

2.12, EXPLICIT EXPRESSIONS FOR d, 8, AND A 77

for the Ch which is equal to 1, that is, if {I*,} (i 7 1, me*, b,(M)) is a base
for the rational p-cycles of M, then

The existence of the v4 is assured by the above theorem. The rp, (i =
1, ..., b,) clearly form a basis for the harmonic forms of degree p and the
fundamental theorem is proved.
Although not explicitly mentioned it should be emphasized that the
existence theorems of de Rham are valid only for orientable manifolds.
The theorem (R,) may be deduced from (R,) and the decomposition
theorem of 5 2.10.

2.12. Explicit expressions for d, 5, and A
In the sequel, unless written otherwise, a p-form a will have the
following equivalent representations:

in the local coordinates ul, -.-, un. We proceed to obtain formulae for
the operators d, 8, and A in a Riemannian manifold-the details of the
computations being left as an exercise. In the first place,

If we write (cf. (1.4.1 1))

then

and