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Chapter 9
Integration on Manifolds
9.1

Integration in Rn

As we said in Section 8.1, one of the raison d’ˆtre for diﬀerential forms is that they are the
e
objects that can be integrated on manifolds. We will be integrating diﬀerential forms that
are at least continuous (in most cases, smooth) and with compact support. In the case of
forms, ω , on Rn , this means that the closure of theset, {x ∈ Rn | ωx ￿= 0}, is compact.
Similarly, for a form, ω ∈ A∗ (M ), where M is a manifod, the support, suppM (ω ), of ω is the
closure of the set, {p ∈ M | ωp ￿= 0}. We let A∗ (M ) denote the set of diﬀerential forms with
c
compact support on M . If M is a smooth manifold of dimension n, our ultimate goal is to
deﬁne a linear operator,
￿
: An (M ) −→ R,
c
M

which generalizes in anatural way the usual integral on Rn .

In this section, we assume that M = Rn , or some open subset of Rn . Now, every n-form
(with compact support) on Rn is given by
ωx = f (x) dx1 ∧ · · · ∧ dxn ,
where f is a smooth function with compact support. Thus, we set
￿
￿
ω=
f (x)dx1 · · · dxn ,
Rn

Rn

where the expression on the right-hand side is the usual Riemann integral of f on Rn .Actually, we will need to integrate smooth forms, ω ∈ An (U ), with compact support deﬁned
c
on some open subset, U ⊆ Rn (with supp(ω ) ⊆ U ). However, this is no problem since we
still have
ωx = f (x) dx1 ∧ · · · ∧ dxn ,

where f : U → R is a smooth function with compact support contained in U and f can be
￿
smoothly extended to Rn by setting it to 0 on Rn − supp(f ). We write V ω forthis integral.
309

310

CHAPTER 9. INTEGRATION ON MANIFOLDS

It is crucial for the generalization of the integral to manifolds to see what the change of
variable formula looks like in terms of diﬀerential forms.
Proposition 9.1 Let ϕ : U → V be a diﬀeomorphism between two open subsets of Rn . If the
Jacobian determinant, J (ϕ)(x), has a constant sign, δ = ±1 on U , then for every ω ∈ An(V ),
c
we have
￿
￿
ϕ∗ ω = δ
ω.
U

V

Proof . We know that ω can be written as

ωx = f (x) dx1 ∧ · · · ∧ dxn ,

x ∈ V,

where f : V → R has compact support. From the example before Proposition 8.6, we have
(ϕ∗ ω )y = f (ϕ(y ))J (ϕ)y dy1 ∧ · · · ∧ dyn
= δ f (ϕ(y ))|J (ϕ)y | dy1 ∧ · · · ∧ dyn .

On the other hand, the change of variable formula (using ϕ) is
￿
￿
f (x) dx1 · ·· dxn =
f (ϕ(y )) |J (ϕ)y | dy1 · · · dyn ,
ϕ(U )

U

so the formula follows.
We will promote the integral on open subsets of Rn to manifolds using partitions of unity.

9.2

Integration on Manifolds

Intuitively, for any n-form, ω ∈ An (M ), on a smooth n-dimensional oriented manifold, M ,
c
￿
the integral, M ω , is computed by patching together the integrals on small-enough opensubsets covering M using a partition of unity. If (U, ϕ) is a chart such that supp(ω ) ⊆ U ,
￿
then the form (ϕ−1 )∗ ω is an n-form on Rn and the integral, ϕ(U ) (ϕ−1 )∗ ω , makes sense. The
orientability of M is needed to ensure that the above integrals have a consistent value on
overlapping charts.
Remark: It is also possible to deﬁne integration on non-orientable manifolds usingdensities
but we have no need for this extra generality.
Proposition 9.2 Let M be a smooth oriented manifold of dimension n. Then, there exists
a unique linear operator,
￿
: An (M ) −→ R,
c
M

with the following property: For any ω ∈ An (M ), if supp(ω ) ⊆ U , where (U, ϕ) is a positively
c
oriented chart, then
￿
￿
( ϕ−1 ) ∗ ω .

ω=

M

ϕ (U )

( †)

9.2. INTEGRATION ON MANIFOLDS311

Proof . First, assume that supp(ω ) ⊆ U , where (U, ϕ) is a positively oriented chart. Then,
we wish to set
￿
￿
ω=
( ϕ−1 ) ∗ ω .
M

ϕ (U )

However, we need to prove that the above expression does not depend on the choice of the
chart. Let (V, ψ ) be another chart such that supp(ω ) ⊆ V . The map, θ = ψ ◦ ϕ−1 , is a
diﬀeomorphism from W = ϕ(U ∩ V ) to W ￿ = ψ (U ∩ V ) and,...