Math 343 Homework 4
Due Wednesday February 24, 2010.
1
To be handed in for grading
1. Suppose that
Z
is an
l
dimensional submanifold of
X
and that
z
∈
Z
. Show that
there exists a local coordinate system
{
x
1
, x
2
, . . . x
k
}
defined in a neighborhod
U
of
z
in
X
such that
Z
∩
U
is defined by the equations
x
l
+1
= 0
, . . . , x
k
= 0. State any
results you use and show clearly why they work.
2. If
f
:
M
→
N
and
g
:
N
→
P
are immersions, show that
g
◦
f
:
M
→
P
is an
immersion as well.
3. Let
f
:
R
3
→
R
be given by
f
(
x, y, z
) =
x
2
+
y
2

z
2
.
(a) Check that 0 is the only critical value of
f
.
(b) Prove that if
a
and
b
are either both positive or both negative, then
f

1
(
a
) and
f

1
(
b
) are diffeomorphic.
(c) Draw pictures to explain what happens to the topology of
f

1
(
c
) as
c
moves
through the criticial value.
4. Let
M
= all 2
×
2 matrices
A
=
x
y
z
w
with real coefficients and det
A
= 1 and
AA
T
diagonal (that is, offdiagonal entries all 0).
(a) Describe
M
as the solution set of 2 equations in 4 variables.
(b) Show
M
is a manifold. State any results you use and show clearly why they can
be used.
5. For which values of
a
does the hyperboloid defined by
x
2
+
y
2

z
2
= 1 intersect the
sphere
x
2
+
y
2
+
z
2
=
a
transversally in
R
3
? What does the intersection look like for
different values of
a
?
1
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2
Extra Practice
1.
(a) If
f
and
g
are immersions, show that
f
×
g
is.
(b) If
f
is an immersion, show that its restriction to any submanifold on its domain
is an immersion.
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 Spring '10
 ElizabethDenne
 Linear Algebra, Vector Space, Manifold, Differential topology, Differentiable manifold

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