Math 8061 Homework 7
Due Tuesday, 11/23/10
1. Do problem 64 of Lee.
2. Do problem 123 of Lee.
3. A kform on a vector space V is called decomposable if it can be expressed as
= 1 k,
for 1forms 1 , . . . , k .
a) Prove that every 2form on R2 and every 2for
Math 8061 Homework 8
Due Monday, 12/13/10
1. Suppose M and N are connected, oriented smooth nmanifolds, and f : M N is an
immersion. Prove that f is orientationpreserving everywhere or orientationreversing everywhere.
2. Let T 2 R4 be a torus, parametrize
Math 8061 Homework 1
Due Thursday, 9/9/10
1. Let M be an ndimensional manifold with boundary. Prove that the boundary M is an
(n 1) dimensional manifold, without boundary.
2. A metric space X is called connected if it cannot be expressed as the disjoint u
Math 8061 Homework 4
Due Thursday, 10/14/10
1. Let C be a circle smoothly embedded in R4 . Prove that there exists a 3dimensional
hyperplane H, such that the orthogonal projection : C H is an embedding. Hint: use
Sards theorem.
2. Do problem 3-2 on page 7
Math 8061 Homework 3
Due Thursday, 9/30/10
1. Let M be a smooth compact nmanifold. Prove that there is no submersion f : M Rk ,
for any k > 0.
2. If M is compact, prove that every 11 immersion f : M N is an embedding.
3. Let RP2 = S 2 / , where antipodal
Math 8061 Homework 2
Due Thursday, 9/23/10
1. Let CPn be the complex projective nspace. That is,
CPn = Cn+1 / ,
where (z0 , . . . , zn ) (z0 , . . . , zn ) for C cfw_0.
Prove that CPn is a smooth manifold, of real dimension 2n. Hint: see page 7 of Lees bo
Math 8061 Homework 6
Due Thursday, 11/11/10
1. Consider the following vector elds on R2 :
V =x
+y ,
y
x
W =x
y .
y
x
Prove that these vector elds do not commute in two ways:
a) by checking that [V, W ] = 0.
b) by computing their ows, and checking that the
Math 8061 Homework 5
Due Thursday, 10/28/10
1. Let f : Rn R be a smooth function, and let a R be a regular value of f . Earlier in class,
we showed that M = f 1 (a) is a smooth (n 1) manifold. Now, prove that M is orientable.
2. Think of S 3 as a subset o