Lecture 33
6.5
Differential
Forms
on
Manifolds
Let
U
⊆
R
n
be
open.
By definition, a
k
form
ω
on
U
is
a
function
which
assigns
to
each
point
p
∈
U
an element
ω
p
∈
Λ
k
(
T
∗
R
n
).
p
We
now
define
the
notion of
a
k
form
on
a
manifold.
Let
X
⊆
R
N
be an
n

dimensional
manifold.
Then,
for
p
∈
X
, the tangent space
T
p
X
⊆
T
p
R
N
.
Definition 6.14.
A
k
form
ω
on
X
is
a
function
on
X
which
assigns
to
each
point
p
∈
X
an element
ω
p
∈
Λ
k
((
T
p
X
)
∗
).
Suppose that
f
:
X
→
R
is
a
C
∞
map, and
let
f
(
p
) =
a
. Then
df
p
is
of the form
df
p
:
T
p
X
T
a
R
∼
R
.
(6.47)
=
→
We
can think of
df
p
∈
(
T
p
X
)
∗
= Λ
1
((
T
p
X
)
∗
). So, we get a
oneform
df
on
X
which
maps each
p
∈
X
to
df
p
.
Now,
suppose
µ
is
a
k
form
on
X
, and
(6.48)
ν
is
an
�
form
on
X
.
(6.49)
For
p
∈
X
,
we
have
µ
p
∈
Λ
k
(
T
∗
X
) and
(6.50)
p
ν
p
∈
Λ
�
(
T
∗
X
)
.
(6.51)
p
Taking
the
wedge
product,
µ
p
∧
ν
p
∈
Λ
k
+
�
(
T
∗
X
)
.
(6.52)
p
The
wedge
product
µ
∧
ν
is
the
(
k
+
�
)form
mapping
p
∈
X
to
µ
p
∧
ν
p
.
Now we
consider
the
pullback operation. Let
X
⊆
R
N
and
Y
⊆
R
�
be manifolds,
and
let
f
:
X
→
Y
be
a
C
∞
map. Let
p
∈
X
and
a
=
f
(
p
). We have the map
df
p
:
T
p
X
T
a
Y.
(6.53)
→
From
this
we
get
the
pullback
(
df
p
)
∗
: Λ
k
(
T
a
∗
Y
)
→
Λ
k
(
T
∗
X
)
.
(6.54)
p
Let
ω
be
a
k
form
on
Y
.
Then
f
∗
ω
is
defined
by
(
f
∗
ω
)
p
= (
df
p
)
∗
ω
q
.
(6.55)
1
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�
Let
f
:
X
→
Y
and
g
:
Y
→
Z
be
C
∞
maps
on
manifolds
X, Y, Z
. Let
ω
be a
k
form. Then
(
g
◦
f
)
∗
ω
=
f
∗
(
g
∗
ω
)
,
(6.56)
where
g
◦
f
:
X
Z
.
→
So
far,
the
treatment
of
k
forms
for manifolds
has
been
basically
the same as
our
earlier
treatment
of
k
forms.
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 Fall '04
 unknown
 Topology, Open set, Euclidean space, Topological space, Ωk

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