Page 34 - Curvature and Homology

16 I. RIEMANNIAN MANIFOLDS

where p, q, and r are functions of class 2 (at least) of x, y, and s hh for
its differential the 2-form

Moreover, the 2-form

has the differential

In more familiar language, da is the curl of a and dp its divergence.
That dda = 0 is expressed by the identity
div curl a = 0.

We now show that the coefficients ails.,, of a differential form u
can be considered as the components of a skew-symmetric tensor field
of type (0,p). Indeed, the a, ,, are defined for i, < ... < 4. They
may be defined for all values of the iadices by taking account of the
anti-commutativity of the covectors dug, that is we may write

That the a81...C are the components of a tensor field is easy to show.
In the sequel, we will absorb the factor l/p! in the expression of a
p-form except when its presence is important.
In order to express the exterior product of two forms and the
differential of a form (cf. (1.4.5)) in a canonical fashion the Kronecker
symbol

will be useful. The important properties of this symbol are:
(i) 82::::: is skew-symmetric in the i, and j,,
(i~--*fp'
$1
.
(i) 8, = i i,
This condition is equivalent to