Page 32 - Curvature and Homology

14 I. RIEMANNIAN MANIFOLDS

Let V* denote the dual space of V and consider the Grassman
algebra A (V*) over V*. It can be shown that the spaces Ap(V*) are
canonically isomorphic with the spaces (AP(V))* dual to Ap(V). The
linear space AP(V*) is called the space of exterior p-forms over V;
its elements are called p-forms. The isomorphism between AP(V*) and
p-p(V*) will be considered in Chapter 11, 5 2.7 as well as in 1I.A.
We return to the vector space T$ of covariant vectors at a point P
of the differentiable manifold M of class k and let U be a coordinate
neighborhood containing P with the local coordinates ul, ..., un and
natural base dul, ..., dun for the space T$. An element a(P) E ~p(Tp*)
then has the following representation in U:

a(P) = a( i,, (P) duil(P) A ... A duip(P). (1.4.4)
If to each point PE U we assign an element a(P) E Ap(T$) in such a way
that the coefficients ail...% are of class 1 2 I (1 < k) then or is said to be a
dt#erential form of degree p and class 1. More precisely, an exterior
dtflerential polynomial of class 1 k - 1 is a cross-section or of class 1
of the bundle
A*(M) = A(T*) = U A (T:),
PEM
that is, if .n is the projection map:

defined by T(A(T$)) = P, then or : M -+ A *(M) must satisfy m(P) = P
for all P E M (cf. 5 1.3 and I. J). If, for every P E M, a(P) E Ap(T$)
for some (fixed) p, the exterior polynomial is called an exterior dz#erential
form of degree p, or simply a p-form. In this case, we shall simply write
or E AP(T*). (When reference to a given point is unnecessary we shall
usually write T and T* for Tp and T,* respectively).
Let M be a differentiable manifold of class k 2 2. Then, there is a map
d : A (T*) -+ A (T*)

sending exterior polynomials of class 1 into exterior polynomials of class
1 - 1 with the properties: .
(i) For p = 0 (differentiable functions f), df is a covector (the
differential off),
(ii) d is a linear map such that d( AP(T*)) C Ap+l(T*),
(iii) For a E Ap(T*), /3 E AQ(T*),