Exam 2: Math 7350 Spring 2011
Problem 1. (10)
Let
V
be a real vector space of dimension
N
. Prove that
dim
(
Σ
k
(
V
)
)
=
±
N
+
k

1
k
²
=
(
N
+
k

1)!
k
!(
N

1)!
.
Problem 2. (10)
Let (
M,g
1
) and (
N,g
2
) be Riemannian manifolds. Suppose
F
:
M
→
N
is a smooth map such
that
F
*
(
g
2
) =
g
1
. Prove that
F
is an immersion.
Problem 3. (10)
Let
N
>
3 be an integer and let
V
be a real vector space of dimension
N
. Prove that Σ
3
(
V
)
⊕
Λ
3
(
V
)
6
=
T
3
(
V
).
Problem 4. (10)
Suppose
M
and
N
are oriented smooth manifolds and
F
:
M
→
N
is a local diﬀeomorphism. If
M
is connected, prove that
F
is either orientationpreserving or orientationreversing.
Problem 5. (10)
Let
ω
be the (
N

1)form on
R
N
\ {
0
}
deﬁned by
ω
=
k
x
k

N
N
X
i
=1
(

1)
i

1
x
i
d
x
1
∧ ··· ∧
d
d
x
i
∧ ··· ∧
d
x
N
,
where ‘hat’ denotes deletion of a term. Prove that
ω
is closed but not exact on
R
N
\ {
0
}
.
Problem 6. (10)
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 Spring '08
 Staff
 Vector Space, Manifold, real vector space, smooth vector field

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