Page 37 - Curvature and Homology

and put

Clearly, +* is a ring homomorphism. Moreover,

that is, the exterior differential operator d commutes with the induced
dual map of a differentiable map from one differentiable manifold into
another.

1.6. Integration of differential forms

It is our intention in this section to sketch a proof of the formula of
Stokes not merely because of its fundamental importance in the theory
of harmonic integrals but because of the applications we make of it
in later chapters. However, a satisfactory integration theory of differential
forms over a differentiable manifold must first be developed.
The classical definition of a p-fold integral

of a continuous function f = f(ul, ..., UP) of p variables defined over
a domain D of the space of the variables ul, ..., up as given, for example,
by Goursat does not take explicit account of the orientation of D. The
definition of an orientable differentiable manifold M given in 5 1.1
together with the isomorphism which exists between Ap(T,*) and
An-p(T,*) at each point P of M (cf. 5 2.7) results in the following
equivalent definition:
A differentiable manifold M of dimension n is said to be orientable
if there exists over M a continuous differential form of degree n which
is nowhere zero (cf. 1.B).
Let a and fi define orientations of M. These orcentations are the same
if /3 =fa where the function f is always positive. An orientable manifold
therefore has exactly two orientations. The manifold is called oriented
if such a form a # 0 is given. The form or induces an orientation in the
tangent space at each point P E M. Any other form of degree n can theh
be written as f(P)a and is be -said to be > 0, < 0 or = 0 at P provided
that f(P) > 0, < 0 or = 0. This depends only on the orientation of M
and not on the choice of the differential form defining the orientation.
The carrier, carr (a) of a differential form or is the closure of the set
of points outside of which or is equal to zero. The following theorem
due to J. Dieudonnk is of crucial importance. (Its proof is given in Appen-
dix D.)