Differential Topology - Computer Graphics and Geometric Modeling - page 536

where p s is a point of the manifold on or close to the cell s, to approximate the inte-
gral. This would be the numerical approach. The limit of such sums could be defined
to be the integral. Unfortunately, this would not work for abstract manifolds. The
approach that works in general and that is also more elegant uses differential forms.
We already defined these for open subsets of Euclidean space. To define them for
manifolds, we use the fact that manifolds look like Euclidean space locally and use
the coordinate neighborhoods of a manifold to carry the definitions we gave for
Euclidean space over to the manifold. For a more thorough discussion than given
here, see [Spiv70a] or [GuiP74].

Note. In this chapter we have given more than one definition for both manifolds
and their tangent vectors. Lack of space prevents us from giving all the corresponding
variants for definitions in this section. We shall therefore assume that all manifolds
are abstract manifolds and use the linear functional approach to tangent vectors
for manifolds. The translation of definition and results for manifolds in R n and for

the equivalence class of vectors approach will be left as exercises.

Let M n be a differentiable manifold (possibly with boundary). Define a vector

k M = ( E ,p, M n ) by

bundle w

U

( ( ) )

k n

E = L T M ,

p

n

pM Œ

and

( ( ) )

n k n

p: EM Æ is the map that sends v Œ L T M to p .

p

The local coordinate charts for w k M as well as the topology for E are defined in a
fashion very similar to what was done for the tangent bundle of M and we shall leave
that as an exercise for the reader. The fibers L k (T p ( M n )) of w k M are just vector spaces
of alternating multilinear maps or exterior k-forms. The total space E of w k M is actu-
ally a differentiable manifold of dimension n +

n
k

Ê
Ë

ˆ
¯

.

n
k

Ê
Ë

ˆ
¯

-dimensional vector bundle w k M is called the exterior k-form

Definition. The

bundle of M .

Other than the definition, the reader does not need to know anything else about
w k M . The reason for introducing w k M is that it provides a convenient way to talk about
differential forms below because the best way to think of a differential k-form is as a
cross-section in that bundle.

Definition. A differential k-form on M , or simply k-form or differential form , is a cross-
section of the bundle w k M , that is, it is a map w defined on M that sends p ΠM to an
element w( p ) ŒL k (T p ( M )). The vector space of differential k-forms on M will be
denoted by W k 0 ( M ). There is a wedge product