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Take Home Final Math 250A Fall 2010
1. Let
M
be a
C
1
manifold and let
X
M
. Let
f
2
C
1
(
M
)
be such that
Xf
= 0
and let
t
be the local ±ow generated by
X
.
Then prove that for every
p
2
M
,
f
(
t
(
p
)) =
f
(
p
)
:
Use this observation
to describe the leaves of the foliation of
R
2
(0
;
0)
g
corresponding to the
subbundle of the tangent bundle given by
p
!
R
X
p
with
X
=
y
2
+
x
2
:
(Hint: Consider
x
3
y
3
:
)
2. Let
f
:
M
!
R
N
be a proper embedding of a noncompact manifold
M
.
Show that
f
(
M
)
cannot be bounded.
3. Use the method of the proof we gave of the embedding theorem for
compact manifolds to prove that a compact
m
dimensional manifold with
boundary can be embedded in
R
2
m
+1
. Use this result to prove that the
interior of a compact
m
dimensional manifold with boundary has a proper
embedding into
R
2
m
+1
.
4. The purpose of this problem is to guide you through a proof of: If
M
is
a connected
C
1
manifold and
p;q
2
M
then there exists a di/eomorphism
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 Fall '10
 N.R.Wallach
 Math

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