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Exam 1: Math 7350 Spring 2011
Problem 1. (15)
(a)
(10)
Let
M
be a smooth manifold and let
S
⊂
M
be an embedded submanifold. Prove that
S
is a closed
subset of
M
if and only if the inclusion map
ι
:
S ,
→
M
is a proper map. (Recall that if
X
and
Y
are
topological spaces and
F
:
X
→
Y
is a map, then
F
is proper if
F

1
(
C
) is compact for every compact set
C
⊂
Y
.)
(b)
(5)
Give an example of a smooth manifold
M
and an embedded submanifold
S
⊂
M
such that
S
is not
closed in
M
.
Problem 2. (10)
Let
M
and
N
be smooth manifolds. If
F
:
M
→
N
is a smooth map and
c
∈
N
is such that
F

1
(
c
) is an embedded submanifold of
M
with codim(
F

1
(
c
)) = dim(
N
), must
c
be a regular value of
F
? Either
prove this or give a counterexample.
Problem 3. (10)
(a)
(5)
Let
M
be a smooth manifold and let
S
⊂
M
be an embedded submanifold. Suppose
γ
:
J
→
M
is a
smooth curve such that
γ
(
J
)
⊂
S
. Prove that for every
t
∈
J
,
γ
0
(
t
) is in the subspace
T
γ
(
t
)
S
of
T
γ
(
t
)
M
.
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This note was uploaded on 10/03/2011 for the course MATH 3364 taught by Professor Staff during the Spring '08 term at University of Houston.
 Spring '08
 Staff
 Logic

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