Page 38 - Curvature and Homology

20 I. RIEMANNIAN MANIFOLDS

To a locally finite open covering {U*} of a differentiable manifold of
class k 2 1 there is associated a set of functions kj} with the properties
(i) Each gj is of class k and satisfies the inequalities

everywhere. Moreover, its carrier is compact and is contained in one
of the open sets U*,

(iii) Every point of M has a neighborhood met by only a finite number
of the carriers of g,.
The gj are said to form a partition of unity subordinated to {U*} that is,
a partition of the function 1 into non-negative functions with small
carriers. Property (iii) states that the partition of unity is locally finite,
that is, each point PE M has a neighborhood met by only a finite number
of the carriers of gj. If M is compact, there can be a finite number of gj;
in any case, the gj form a countable set. With these preparations we can
now prove the following theorem:
Let M be an oriented differentiable manifold of dimension n. Then,
there is a unique functional which associates to a continuous differential
form a of degree n with compact carrier a real number denoted by JMa
and called the integral of a. This functional has the properties:

(ii) If the carrier of a is contained in a coordinate neighborhood U
with the local coordinates ul, ..., un such that dul A ... A dun > 0 (in U)
is
and a = a,..., dul A ... A dun where ct ,... , a function of ul, ..., un, then

where the n-fold integral on the right is a Riemann integral.
Since carr (a) c U we can extend the definition of the function a,.., to
the whole of En, so that (1.6.1) becomes the the n-fold integral

In order to define the integral of an n-form a with compact carrier S
we take a locally finite open covering {U4} of M by coordinate neighbor-
hoods and a partition of unity {gj} subordinated to {U*}. Since every
point P E S has a neighborhood met by only a finite number of the