6Integration
III Differential Geometry
6.1 Orientation
We start with the notion of an orientation of a vector space. After we have one,
we can define an orientation of a manifold to be a smooth choice of orientation
for each tangent space.
Informally, an orientation on a vector space
V
is a choice of a collection of
ordered bases that we declare to be “oriented”. If (
e
1
, · · · , e
n
) is an oriented
basis, then changing the sign of one of the
e
i
changes orientation, while scaling
by a positive multiple does not. Similarly, swapping two elements in the basis
will induce a change in orientation.
To encode this information, we come up with some alternating form
ω ∈
Λ
n
(
V
∗
). We can then say a basis
e
1
, · · · , e
n
is oriented if
ω
(
e
1
, · · · , e
n
) is positive.
Definition
(Orientation of vector space)
.
Let
V
be a vector space with
dim V
=
n
. An orientation is an equivalence class of elements
ω ∈
Λ
n
(
V
∗
), where we say
ω ∼ ω
0
iff ω = λω
0
for some λ > 0. A basis (e
1
, · · · , e
n
) is oriented if
ω(e
1
, · · · , e
n
) > 0.
By convention, if V = {0}, an orientation is just a choice of number in {±1}.
Suppose we have picked an oriented basis
e
1
, · · · , e
n
. If we have any other
basis ˜e
1
, · · · , ˜e
n
, we write
e
i
=
X
j
B
ij
˜e
j
.
Then we have
ω(˜e
1
, · · · , ˜e
n
) = det B ω(e
1
, · · · , e
n
).
So ˜e
1
, · · · , ˜e
n
is oriented iff det B > 0.
We now generalize this to manifolds, where we try to orient the tangent
bundle smoothly.
Definition
(Orientation of a manifold)
.
An orientation of a manifold
M
is
defined to be an equivalence class of elements
ω ∈
Ω
n
(
M
) that are nowhere
vanishing, under the equivalence relation
ω ∼ ω
0
if there is some smooth
f
:
M → R
>0
such that ω = fω
0
.
Definition
(Orientable manifold)
.
A manifold is orientable if it has some
orientation.
If
M
is a connected, orientable manifold, then it has precisely two possible
orientations.
Definition
(Oriented manifold)
.
An oriented manifold is a manifold with a
choice of orientation.
Definition
(Oriented coordinates)
.
Let
M
be an oriented manifold. We say
coordinates x
1
, · · · , x
n
on a chart U are oriented coordinates if
∂
∂x
1
p
, · · · ,
∂
∂x
n
p
is an oriented basis for T
p
M for all p ∈ U .
Note that we can always find enough oriented coordinates. Given any
connected chart, either the chart is oriented, or
−x
1
, · · · , x
n
is oriented. So any
oriented M is covered by oriented charts.
Now by the previous discussion, we know that if
x
1
, · · · , x
n
and
y
1
, · · · , y
n
are oriented charts, then the transition maps for the tangent space all have
positive determinant.
Example. R
n
is always assumed to have the standard orientation given by
dx
1
∧ · · · ∧ dx
n
.
Definition
(Orientation-preserving diffeomorphism)
.
Let
M, N
be oriented
manifolds, and
F ∈ C
∞
(
M, N
) be a diffeomorphism. We say
F
preserves
orientation if D
F |
p
:
T
p
M → T
F (p)
N
takes an oriented basis to an oriented
basis.
Alternatively, this says the pullback of the orientation on
N
is the orientation
on M (up to equivalence).