Page 40 - Curvature and Homology

22 I. RIEMANNIAN MANIFOLDS

Although the form ha is not continuous the right side is meaningful
as one sees by taking a partition of unity.
Let a be a differential form of degree n - 1 and class k 2 1 in M.
Then

where the map i sending aD into M is the identity and aD has the
orientation canonically induced by that of D. This is the theorem of
Stoker. In order to prove it, we select a countable open covering of M
by coordinate neighborhoods {U,) in such a way that either UZ does
not meet aD, or it has the property of the neighborhood U in the
definition of boundary point. Let kj) be a partition of unity subordinated
to this covering. Since D and its boundary are both compact, each of
them meets only a finite number of the carriers of gj. Hence,

and

These sums being finite, it is only necessary to establish that

for each i, the integrals being evaluated by f~rmula (1.6.1). To complete
the proof then, choose a local coordinate system ul, ..., un for the
coordinate neighborhood Ui in such a way that dul A ... A dun > 0
and put

where the functions a, are of class 2 1. Then,

Compare with (1.4.9). The remainder of the proof is left as an exercise.