Page 90 - Curvature and Homology

11. TOPOLOGY OF DIFFERENTIABLE MANIFOLDS

and
Ei = D,f,
locally.
Moreover,
Dip=()
and
Bij = DjA, - DiAj

where the skew-symmetric tensor field

In the language of differential forms, if we denote by 7 and a the
1-forms defined by E and A and by fi the 2-form defined by the
bivector Bij, then the equations (2.8.2) - (2.8.5) become

p
- = da (locally).
We note that fi = *p where p is the 1-form corresponding to the
covariant vector field gijBj where gij is the metric tensor of 8.
Now, the theorems of classical potential theory, namely, (a) if 7 is
closed, then 7 is exact and (b) if is closed, then fi is exact are not
necessarily true in an arbitrary 3-dimensional differentiable manifold
since the first and second betti numbers may not vanish (cf. 5 2.4).
We digress for a moment and consider a Riemannian manifold of
dimension n. To a p-form or on M we associate a (p - 1)-form 8or
defined in terms of the operators d and *:

The form 8a ig called the co-dzj(gerentia1 of a and has the properties:
(i) 8(or + ;8) = Sor + sfi,
(ii) 88or = 0,
(iii) *8or = (- l)P dm, *da = (- l)p+l 8 *or.
The form a is said to be co-closed if its co-differential is zero. This is
equivalent to the statement that its adjoint is closed. If or = 8fi we say
that or coboundi and that or is co-exact;
It should be remarked that in contrast with the differential operator d,