Page 91 - Curvature and Homology

the co-differential operator 6 involves the metric structure of M in an
essential way.
A form a is said to be harmonic (or a harmonic field) if it is closed and
co-closed. This is the definition given by Hodge. K. Kodaira [4q, on
the other hand calls a form ar harmonic if Aa = 0 where A is the
(Laplace-Beltrami) operator d8 + Sd. It is evident that the harmonic
forms of a given degree form a linear space. However, since the operator
A is not, in general, a derivation, they do not form an algebra.
If ar is the form of degree 1 in E3 associated with the vector V, then
the forms d8a and 8da are associated with the vectors grad div V and
curl curl V and hence the form Aa is associated with the vector field
VZ V = grad div V - curl curl V. Now, in the above example, at any
point of E3 where there is no current, the vector potential A satisfies
the equation curl curl A = 0. Regarding the vector field E, the 1-form
associated with it is harmonic, and so from the equation (2.8.1) we
conclude that the potential difference between two points in an electrical
field is given by the integral of the harmonic form 77 along 'any' path
connecting the points. Moreover, the integral &A dr of the vector
potential in the magnetic field round a bounding cycle r is equal to
the integral of the 2-form #I over 'any' 2-chain C with r = aC, that is,

In§ 2.10 we shall sketch a proof of the statement that there are harmonic
p-fa (0 < p < n) on an n-dimensional Riemannian manifold M with
the property that the integral

has mbitrariZy prescribed periods on b,(M) independent p-cycles of M.
This generalizes the above results for the forms 7 and #I.

2.9. Orthogonality relations
We shall assume in the remaining sections of this chapter that the
Riemannian manifold M is compact and orientable. Let or and /I be
forms of degree p and p + 1, respectively. Then, by Stokes' theorem