§ 4. Differential Manifolds
263
as in (9.20). The invariance of the integrals of ω under homotopy now follow
if ω isclosed. Notice that ω is an exact differential if and only if its integral
along every closed path is zero. Etc.
A much more serious problem consists in defining the extended integral
of a form of maximum degree on X without taking a “ parametric representation ” of X for granted for otherwise we would only need to integrate over
a cube.
(i) Orientable manifolds. Let X be an n-dimensional manifold and ω a
differential form of degree n on X and of class C
0 . Whatever be the final
definition of the integral of ω, if X is not compact, we will clearly come
across convergence problems at infinity unrelated to the problem at hand.
This is already obvious if X = R. It is, therefore, prudent to suppose that ω
has compact support
66 in order to get rid of them, as we did at the start of
Chapter V.
The simplest case is obtained by supposing the existence of a chart for all
of X, in other words, that X is diffeomorphic to an open subset of a Cartesian
space. For this, let us choose a diffeomorphism ϕ from X onto ϕ(X) ⊂ R
n ;
it transforms ω into a form
ω(ϕ) = a(ξ)dξ
1
∧ . . . ∧ dξ
n ,
on ϕ(X) and we wish to set
X
ω =
ϕ(X)
a(ξ)dξ
i . . . dξ
n .
(17.1)
This a an ordinary multiple integral over the open set ϕ(X) of a function
which vanishes outside a compact subset of it.
If ψ is another diffeomorphism from X onto an open subset of R
n , then
there is a form
ω(ψ) = b(η)dη
1
∧ . . . ∧ dη
n
on ψ(X). As ω(ϕ) is the inverse image of ω(ψ) under the chart change diffeomorphism θ : ϕ(X) −→ ψ(X)
a(ξ) = b [θ(ξ)] J θ (ξ) .
Since the change of variable formula for multiple integrals showing that
ψ(X)
b(η)dη
1 . . . dη
n =
ϕ(X)
b [θ(ξ)] |J θ (ξ)|dξ
1 . . . dξ
n ,
we are led to conclude that the two values proposed for the integral of ω are
equal only if J θ (ξ) > 0 everywhere. This is not a good sign in every sense of
the word since we want a result independent of the coordinate system used.
66 Recall that it is the largest closed set Supp(. . .) outside which the function (or
differential form, or. . . ) is zero.
263
as in (9.20). The invariance of the integrals of ω under homotopy now follow
if ω isclosed. Notice that ω is an exact differential if and only if its integral
along every closed path is zero. Etc.
A much more serious problem consists in defining the extended integral
of a form of maximum degree on X without taking a “ parametric representation ” of X for granted for otherwise we would only need to integrate over
a cube.
(i) Orientable manifolds. Let X be an n-dimensional manifold and ω a
differential form of degree n on X and of class C
0 . Whatever be the final
definition of the integral of ω, if X is not compact, we will clearly come
across convergence problems at infinity unrelated to the problem at hand.
This is already obvious if X = R. It is, therefore, prudent to suppose that ω
has compact support
66 in order to get rid of them, as we did at the start of
Chapter V.
The simplest case is obtained by supposing the existence of a chart for all
of X, in other words, that X is diffeomorphic to an open subset of a Cartesian
space. For this, let us choose a diffeomorphism ϕ from X onto ϕ(X) ⊂ R
n ;
it transforms ω into a form
ω(ϕ) = a(ξ)dξ
1
∧ . . . ∧ dξ
n ,
on ϕ(X) and we wish to set
X
ω =
ϕ(X)
a(ξ)dξ
i . . . dξ
n .
(17.1)
This a an ordinary multiple integral over the open set ϕ(X) of a function
which vanishes outside a compact subset of it.
If ψ is another diffeomorphism from X onto an open subset of R
n , then
there is a form
ω(ψ) = b(η)dη
1
∧ . . . ∧ dη
n
on ψ(X). As ω(ϕ) is the inverse image of ω(ψ) under the chart change diffeomorphism θ : ϕ(X) −→ ψ(X)
a(ξ) = b [θ(ξ)] J θ (ξ) .
Since the change of variable formula for multiple integrals showing that
ψ(X)
b(η)dη
1 . . . dη
n =
ϕ(X)
b [θ(ξ)] |J θ (ξ)|dξ
1 . . . dξ
n ,
we are led to conclude that the two values proposed for the integral of ω are
equal only if J θ (ξ) > 0 everywhere. This is not a good sign in every sense of
the word since we want a result independent of the coordinate system used.
66 Recall that it is the largest closed set Supp(. . .) outside which the function (or
differential form, or. . . ) is zero.