Differential Geometry 1
Homework 05
1. Let (
x, y, z
) be the standard coordinates on
R
3
and (
X, Y
) those on
R
2
.
Let
ω
∈
Ω
2
(
S
2
) be the standard twoform:
ω
=
x
d
y
∧
d
z
+
y
d
z
∧
d
x
+
z
d
x
∧
d
y
and let
F
:
R
2
→
S
2
be inverse to stereographic projection from the north
pole (0
,
0
,
1). Prove that
F
*
ω
=

4
(
X
2
+
Y
2
+ 1)
2
d
X
∧
d
Y.
2. Prove explicitly that the twosphere has trivial de Rham cohomology in
degree one:
H
1
dR
(
S
2
) = 0
.
Suggestion
: Consider stereographic projection from the N and S poles.
Solutions
(1) In terms of the coordinates as given, note that
x
◦
F
=
2
X
R
2
+ 1
,
y
◦
F
=
2
Y
R
2
+ 1
,
z
◦
F
=
R
2

1
R
2
+ 1
where
R
2
=
X
2
+
Y
2
. It follows that
F
*
(d
x
) = d(
x
◦
F
) =
2
(
R
2
+ 1)
2
(1 +
Y
2

X
2
)d
X

2
XY
d
Y
F
*
(d
y
) = d(
y
◦
F
) =
2
(
R
2
+ 1)
2
(1 +
X
2

Y
2
)d
Y

2
Y X
d
X
F
*
(d
z
) = d(
z
◦
F
) =
4
(
R
2
+ 1)
2
X
d
X
+
Y
d
Y
.
Writing
S
=
R
2
+ 1 for further convenience, it follows that
F
*
(
x
d
y
∧
d
z
) =
16
S
5
X

2
XY
2

(1 +
X
2

Y
2
)
X
d
X
∧
d
Y
1
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or
F
*
(
x
d
y
∧
d
z
) =

16
X
2
S
4
d
X
∧
d
Y
and likewise
F
*
(
y
d
z
∧
d
x
) =

16
Y
2
S
4
d
X
∧
d
Y
while
F
*
(
z
d
x
∧
d
y
) =
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 Spring '09
 Robinson
 Ω, Hdr, dx ∧ dy

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