Orientation and Integration
Arshay Sheth
November 29, 2020
Differential forms are objects that are supposed to be integrated on manifolds. In this report,
we set up the machinery that will enable us to do this.
1
Integration of n-forms on Rn
n
If f is a differentiable
R function with compact support
P on R , recall that its Riemann integral
is defined to be: Rn f |dx1 · · · dxn | := lim|σj |→0 j f (xj )|σj |, where σj are n-dimensional
cubes which cover supp f , xj is a point on σj and |σj | is the volume of σj . Note that, unlike
the usual calculus notation, we put an absolute value sign in the integrand to differentiate
it from the integral of a differential form which we now proceed to define.
Definition 1.1. Let x1 , . . . , xn be co-ordinates on Rn and let ω = f dx1 ∧ · · · ∧ dxn be an
n-form with compact support on Rn , where f ∈ Cc∞ (Rn ). Its integral over Rn is defined to
be the Riemann integral of f :
Z
Z
ω :=
f |dx1 · · · dxn |.
Rn
Rn
We now check how the integral of an n-form on Rn transforms with a change of variables. A
change of variables on Rn is a diffeomorphism T = (T1 , . . . , Tn ) : Rn → Rn , with co-ordinates
i
y1 , . . . , yn and x1 , . . . , xn respectively. Letting J(T ) = det( ∂T
)
be the Jacobian deter∂yj 1≤i,j≤n
minant, we first note that
∂T1
∂T1
∂Tn
∂Tn
dy1 + · · · +
dyn ) ∧ · · · ∧ (
dy1 + · · · +
dyn )
∂y1
∂yn
∂y1
∂yn
X
∂Tn
∂T1
=
sign(σ)
···
dy1 ∧ · · · ∧ dyn
∂yσ(1)
∂yσ(n)
σ∈S
dT1 ∧ · · · ∧ dTn = (
n
= J(T )dy1 ∧ · · · ∧ dyn .
1
We thus have that:
T ∗ ω = (f ◦ T )dT1 ∧ · · · ∧ dTn
(using definition of pullback of differential forms)
= (f ◦ T )J(T )dy1 ∧ · · · ∧ dyn
(from above)
and so
Z
Z
∗
(f ◦ T )J(T )|dy1 · · · dyn |
T ω=
Rn
(from Definition 1.1).
Rn
On the other hand, using the change of variables formula from multivariable calculus,
Z
Z
f |dx1 · · · dxn | =
ω=
.
Rn
Z
(f ◦ T )|J(T )| |dy1 · · · dyn |.
Rn
Rn
Thus,
Z
Z
∗
T ω=±
Rn
ω
Rn
depending upon whether J(T ) is positive or negative. This leads us to make the following
definition:
Definition 1.2. A diffeomorphism T : U → V , where U and V are open subsets of Rn is
said to be orientation preserving if J(T ) is everywhere positive.
Thus, the upshot of this section is that the integral of a differential n-form on Rn is only
invariant under orientation preserving diffeomorphisms.
2
Orientation of manifolds
In this section, we define define the notion of an orientation of a manifold and also give an
alternative characterization of this notion. We stick to the convention of reference [1] that
all charts of a manifold have codomain Rn and that for all charts, Rn has fixed co-ordinates
x1 , . . . , xn . Our work in the previous section suggests the following definition is quite natural:
Definition 2.1. Let M be a differentiable manifold with atlas {(Uα , φα )}. We say that this
atlas is oriented if all transition functions gαβ = φα ◦ φ−1
β are orientation preserving. We say
that M is orientable if M has an oriented atlas.
Remark 2.2. We are already intuitively familiar with the idea of orientation: the orientations on R1 are ‘left’ and ‘right’, the orientations on R2 are ‘clockwise’ and ‘anticlockwise’, the orientations on R3 are ‘left-handed’ and ‘right-handed’. However, if our
manifold M = S 1 , we cannot talk about ‘clockwise’ and ‘anticlockwise’ without embdedding
S 1 into R2 . Thus, the utility of our definition of orientation is that it is intrinsic to M and
does not depend on any embedding of M into Euclidean space.
2
Example 2.3 (Orientable manifolds). Euclidean space Rn with the atlas {(Rn , idRn )} is
orientable. The n-sphere S n and the complex projective space CPn are also orientable for
any n ≥ 0.
Example 2.4 (Non-orientable manifolds). The Mobius strip, the Klein bottle and the real
projective space RPn for n even are non-orientable manifolds.
Definition 2.5. A global differential form ω = (ωα )α ∈ Ω(M ) is called an n-form if each
ωα = fα dx1 ∧ · · · ∧ dxn ∈ Ωn (φα (Uα )) = Ωn (Rn ). Moreover, it is called non-vanishing if each
fα ∈ C ∞ (Rn ) is everywhere non zero.
Theorem 2.6. A manifold M is orientable if and only if it has a global non-vanishing
n-form.
Proof. Suppose M has a global non-vanishing n-form ω. Let {(Uα , φα )} be an atlas of
M ; we will use ω to modify this atlas and obtain an oriented atlas. On Uα , write ωα =
fα dx1 ∧· · ·∧dxn . Since ω is nowhere vanishing , fα is either everywhere positive or everywhere
negative on Rn . If fα is everywhere positive on Rn , we leave this chart unchanged. If fα
is everywhere negative on Rn , we replace the chart φα by ψα , where ψα = T ◦ φα and
T : Rn → Rn is a diffeomorphism defined by T (x1 , x2 , . . . , xn ) = (−x1 , x2 , . . . , xn ). To
compute the local representation of ω in this new chart, we use the calculation in Section 1
to note that T∗ (ωα ) = T −1∗ (ωα ) = −(fα ◦ T −1 )dx1 ∧ · · · ∧ dxn . Thus, by abuse of notation,
we will henceforth assume that we have an atlas {(Uα , φα )} such that on each Uα , ωα =
fα dx1 ∧ · · · ∧ dxn with fα everywhere positive on Rn .
Now for any α and β we consider the transition function gαβ = φα ◦ φ−1
β : φβ (Uα ∩ Uβ ) →
∗
φα (Uα ∩ Uβ ). Then gαβ (fα dx1 ∧ · · · ∧ dxn ) = fβ dx1 ∧ · · · ∧ dxn . On the other hand, using the
∗
calculation in Section 1 again, we also know that gαβ
(fα dx1 ∧· · ·∧dxn ) = (fα ◦gαβ )J(gαβ )dx1 ∧
· · · ∧ dxn . Thus, fβ dx1 ∧ · · · ∧ dxn = (fα ◦ gαβ )J(gαβ )dx1 ∧ · · · ∧ dxn . Since fα and fβ are
everywhere positive on Rn , J(gαβ ) is also everywhere positive on Rn . Thus, {(Uα , φα )} is an
oriented atlas for M .
Conversely, suppose {(Uα , φα )} is an oriented atlas for M . Again, we denote gαβ = φα ◦ φ−1
β :
∗
φβ (Uα ∩ Uβ ) → φα (Uα ∩ Uβ ). Then gαβ (dx1 ∧ · · · ∧ dxn ) = J(gαβ )dx1 ∧ · · · ∧ dxn . But we
also have that
∗
∗
∗
gαβ
(dx1 ∧ · · · ∧ dxn ) = (φα ◦ φβ−1 )∗ (dx1 ∧ · · · ∧ dxn ) = φ−1
β (φα (dx1 ∧ · · · ∧ dxn )).
Thus,
φ∗α (dx1 ∧ · · · ∧ dxn ) = φ∗β (J(gαβ )dx1 ∧ · · · ∧ dxn ) = (J(gαβ ) ◦ φβ )φ∗β (dx1 ∧ · · · ∧ dxn ).
Let ωα = φ∗α (dx1 ∧ · · · ∧ dxn ) ∈ Ωn (Uα ) and let ωβ = φ∗α (dx1 ∧ · · · ∧ dxn ) ∈ Ωn (Uβ ) and so
ωα = (J(gαβ ) ◦ φβ )ωβ .
P
Let {ρ
}
be
a
partition
of
unity
subordinate
to
U
.
Define
ω
=
α
α
α ρα ωα . Then on each Uβ ,
P
ω = α ρα (J(gαβ ) ◦ φβ )ωβ . Since each J(gαβ ) is positive function, the ρα ’s are non-negative
functions and not all ρα ’s can vanish at a point, ω is a global non-vanishing n-form.
3
Now suppose ω and ω 0 are two global non-vanishing n-forms on an orientable manifold. On
each Uα , ωα = fα dx1 ∧ · · · ∧ dxn and ωα0 = fα0 dx1 ∧ · · · ∧ dxn where fα and fα0 are nowhere
vanishing smooth functions on Rn . Thus, on Uα we have that ω = ffα0 ω 0 . Using the formula
α
in Section 1 once again, we see that
fβ
(fα ◦ gαβ )J(gαβ )
(fα ◦ gαβ )
= 0
= 0
0
fβ
(fα ◦ gαβ )J(gαβ )
(fα ◦ gαβ )
and so
fβ
fα
|
=
|φ (U ∩U ) .
φ
(U
∩U
)
α
β
β
fβ0
fα0 α α β
Since C ∞ (M ) is a sheaf, we obtain a function f ∈ C ∞ (M ) such that ω = f ω 0 . If M is
connected, then f is everywhere positive or everywhere negative. We define an equivalence
relation on the set of nowhere vanishing global forms by saying that ω and ω 0 are equivalent
if f is positive. Thus, on a connected orientable manifold M , the nowhere vanishing n-forms
fall into two equivalence classes. Each class is called an orientation on M and is denoted
by [M ]. Equipped with this notion, we define integration on oriented manifolds in the next
section.
3
Integration of differential forms on oriented manifolds
Let M be an oriented manifold of dimension n with an oriented atlas {(Uα , φα )}. Let
ω ∈ Ωnc (M ) be any n-form with compact support. First, suppose that supp ω ⊆ Uα for some
α.
We define the integral of ω on M by defining
Z
Z
∗
(φ−1
ω :=
α ) ω.
M
Rn
If (Uβ , φβ ) is another chart such that supp ω ⊆ Uβ then
gαβ = φα ◦ φ−1
β : φβ (Uα ∩ Uβ ) → φα (Uα ∩ Uβ )
is an orientation preserving diffeomorphism and so
Z
Z
Z
−1 ∗
−1 ∗
∗
−1 ∗
(φα ) ω =
(φα ) ω =
(φα ◦ φ−1
β ) (φα ) ω
n
R
φα (Uα ∩Uβ )
φ (U ∩U )
Z β α β
∗
=
(φ−1
α ◦ φα ◦ φβ ) ω
φ (U ∩U )
Z
Z β α β
−1 ∗
∗
=
(φβ ) ω =
(φ−1
β ) ω.
φβ (Uα ∩Uβ )
Rn
Hence, our definition for such ω does not depend on the choice of open set containing supp ω.
4
For the general case, now consider any n-form ω ∈ Ωnc (M ) with compact support. Let
{(Ui , φi )} be a finite cover by oriented charts of supp w. Choose a partition of unity {ρi }
subordinate to the cover {Ui }. Define the integral of ω over M to be
Z
XZ
ω :=
ρi ω.
M
Ui
i
We need to check several things to make sure that this is well-defined:
• Firstly, supp ρi ω ⊆ supp ρi ∩ supp ω ⊆ Ui . Also, since supp ρi ω ⊆ supp ρi ∩ supp ω ⊆
supp w, supp ρi ω is a closed subset of supp ω. Since supp ω is compact by assumption
and since a closed subset of a compact
Hausdorff space is compact, we conclude that
R
supp ρi ω is also compact. Hence, Ui ρi ω is well defined.
• Finally, we check that this definition is independent of both the choice of the open
covering as well as the choice of the partition of unity. Let {(Vj , φ̃j )} be another finite
collection of oriented smooth charts which cover supp ω and let {χj } be a partition of
unity subordinate to {Vj }.
We have that:
XZ
i
ρi ω =
Ui
XZ
i
=
Ui
X
χj = 1)
j
(by linearity of the integral on Rn )
χj ρi ω
Ui
j
XXZ
i
(since
j
XXZ
i
=
X
(
χj )ρi ω
(since supp χj ρi ω ⊆ Ui ∩ Vj ).
χj ρi ω
Ui ∩Vj
j
A completely symmetric argument shows that
XZ
XXZ
χj ω =
j
Vj
j
i
ρi χj ω.
Vj ∩Ui
Since these are finite sums, we conclude that
XZ
XZ
ρi ω =
χj ω
i
Ui
j
Vj
and hence our definition is indeed independent of the choices of the oriented atlas and
the partition of unity.
This completes the set up to define integration on manifolds.
5
We end this section by giving some properties of integration. The proofs of these properties
involve manipulations with definitions and will not be given here.
Theorem 3.1. If M and N are oriented n-manifolds and ω and η are compactly supported
n-forms on Rn , we have the following:
1. (Linearity) If a, b ∈ R, then
Z
Z
aω + bη = a
M
Z
ω+b
M
η.
M
2. (Orientation reversal) If M denotes M with the opposite orientation, then
Z
Z
ω.
ω=−
M
M
3. (Positivity) If the orientation of ω coincides with that of M , then
Z
ω > 0.
M
4. (Diffeomorphism invariance) If f : M → N is an orientation preserving diffeomorphism, then
Z
Z
ω=
f ∗ ω.
M
N
Proof. See [Lee, Proposition 14.6].
4
Orientation on manifolds with boundary
In this section, we introduce the notion of manifolds with boundary and discuss orientability
for such manifolds.
Let Hn := {(x1 , . . . , xn ) ∈ Rn |xn ≥ 0} denote the upper-half space. A continuous map
ϕ : U → V between two open sets U and V of Hn is called a C ∞ -map if it can be extended
to a C ∞ map from an open subset U 0 of Rn containing U to an open subset V 0 of Rn
containing V .
Definition 4.1. A Hausdorff second countable space M is called a manifold with boundary
if there exists an open covering {Uα } of M and homeomorphisms φα : Uα → Vα , where Vα is
either Rn or Hn with the condition that the transition maps gαβ = φα ◦ φ−1
β : φβ (Uα ∩ Uβ ) →
φα (Uα ∩ Uβ ) are C ∞ .
Example 4.2. The upper-half space Hn and the closed disk Dn = {x ∈ Rn |x21 +· · ·+x2n ≤ 1}
are manifolds with boundary.
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We set ∂Hn := {(x1 , . . . , xn ) ∈ Hn |xn = 0} which is homeomorphic to Rn−1 . If M is a
manifold with boundary, we define the boundary of M to be ∂M := {p ∈ M |φα (p) ∈
∂Hn for some α}. Many notions from the theory of manifolds have corresponding counterparts in the theory of manifolds with boundary. For instance, the dimension of a manifold
with boundary and the notion of an oriented manifold with boundary can be defined in
exactly the same way as for manifolds. We now prove an important result that allows us to
relate notions on manifolds with boundary to manifolds.
Theorem 4.3. Let T : Hn → Hn be a diffeomorphism with everywhere positive Jacobian
determinant. Then T induces a map T from ∂Hn to itself. The induced map T viewed as a
diffeomorphism of Rn−1 also has everywhere positive determinant.
Proof. For ease of notation, we will prove the case n = 2; the general case is very similar.
Denote T : H2 → H2 by T = (T1 , T2 ) and let y1 and y2 be the co-ordinates for the first H2 .
By the inverse function theorem, an interior point of H2 must be the image of an interior
point of H2 . Since T is a diffeomorphism, we thus get an induced diffeomorphism:
T : ∂H2 → ∂H2 , (y1 , 0) 7→ (T1 (y1 , 0), 0).
Pick any (a, 0) ∈ ∂H2 . By assumption,
det
∂T1 (a,0)
∂y1
∂T2 (a,0)
∂y1
∂T1 (a,0)
∂y2
∂T2 (a,0)
∂y2
!
>0
(∗).
Since T2 (b, 0) = 0 for all b ∈ R,
T2 (a + h, 0) − T2 (a, 0)
∂T2 (a, 0)
= lim
=0
h→0
∂y1
h
(∗∗).
Since T2 (b1 , b2 ) > 0 for all b2 ∈ R>0 ,
T2 (a, h) − T2 (a, 0)
∂T2 (a, 0)
T2 (a, h)
= lim
= lim
≥ 0.
h→0
h→0
∂y2
h
h
Using (*) and (**), we conclude that
∂T2 (a,0)
∂y2
> 0 and thus
∂T1 (a,0)
∂y1
> 0 as desired.
The theorem implies that if M is a manifold with boundary of dimension n, then ∂M is
naturally an (n − 1)-dimensional manifold. The theorem also implies that an oriented atlas
for M induces an oriented atlas for ∂M ; thus if M is an oriented manifold with boundary,
then ∂M is also oriented manifold.
Let the upper-half space Hn be given the standard orientation dx1 ∧ · · · ∧ dxn . We define the
induced orientation on the boundary ∂Hn to be the equivalence class of (−1)n dx1 ∧ · · · ∧ dxn
for n ≥ 2 and −1 for n = 1. The reason for this seemingly arbitrary choice will become
clear in the next talk. In general, for an oriented manifold M with boundary, we define
the induced orientation [∂M ] on ∂M by the following requirement: if φ is an orientation
preserving diffeomorphism of some open set U in M into Hn , then
φ∗ [∂Hn ] = [∂M ]|∂U ,
where ∂U = ∂M ∩ U .
7
5
Stokes’ theorem
We now have all the ingredients to state the celebrated Stokes’ theorem, which is an important result in the theory of integration on manifolds and the subject of the next talk.
Theorem 5.1 (Stokes’ theorem). If ω is an (n − 1)-form of compact support on an oriented
manifold M of dimension n and ∂M is the induced orientation, then
Z
Z
dω =
ω.
M
∂M
If we set M = [a, b], then ∂M = {a} ∪ {b}. Hence if f is a differentiable function on [a, b]
Rb
(i.e. a 0-form on M ), then Stokes’ theorem translates to a f 0 (x)dx = f (b) − f (a). Hence,
the Fundamental Theorem of Calculus can be regarded as a special case of Stokes’ theorem.
The proof, further corollaries and applications of Stokes’ theorem will be given in the next
talk.
8
References
[1] R. Bott, L.Tu, Differential Forms in Algebraic Topology, Springer-Verlag New York Inc.,
New York, 1982.
[2] J. Lee, Introduction to Smooth Manifolds, Springer-Verlag New York Inc., New York,
2012.
[3] S.Morita, Geometry of Differential Forms, American Mathematical Society, Providence,
2001.
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