- Math 518 Differentiable Manifolds I
- Fall 2013 Final Sample Exam
(Solutions will be posted on the web page on Sunday, December 15)
1. (40 Points) Consider the distribution on M = R3 cfw_(0, 0, z ) : z R
generated by the following 2 vector elds:
X=x
y
y
- Math 518 Differentiable Manifolds I
- Fall 2013 Homework #9
(to be returned at the beginning of class, Friday, Nov. 1)
Read Section 9 from the Lecture Notes and solve the following exercises
from the Lecture Notes:
8.5) Show that any smooth action G M M
- Math 518 Differentiable Manifolds I
- Fall 2013 Homework #7
(to be returned at the beginning of class, Friday, Oct. 18)
Solve the following exercises from the Lecture Notes:
6.5) Let : M N be a smooth map which is transversal to a submanifold Q N (not n
- Math 518 Differentiable Manifolds I
- Fall 2013 Homework #3
(to be returned at the beginning of class, Friday, Sep. 20)
Solve the following exercises from the Lecture Notes:
3.1) Show that f : R R, dened by f (x) = exp(1/x2 ) is a smooth
function.
3.3)
- Math 518 Differentiable Manifolds I
- Fall 2013 Homework #2
(to be returned at the beginning of class, Friday, Sep. 13)
Solve the following exercises from the Lecture Notes:
1.8) Let M be a set and assume that one has a collection C = cfw_(U , ) :
A, w
- Math 518 Differentiable Manifolds I
- Fall 2013 Homework #4
(to be returned at the beginning of class, Friday, Sep. 27)
Solve the following exercises from the Lecture Notes:
4.5) Show that T M has a smooth structure of manifold of dimension
2 dim M , fo
- Math 518 Differentiable Manifolds I
- Fall 2013 Homework #10
(to be returned at the beginning of class, Friday, Nov. 8)
Read Section 10 from the Lecture Notes and solve the following exercises
from the Lecture Notes:
10.1) Complete the computation of th
- Math 518 Differentiable Manifolds I
- Fall 2013 Homework #8
(to be returned at the beginning of class, Friday, Oct. 25)
Solve the following exercises from the Lecture Notes:
(1)
(2)
7.4) Let F1 = cfw_L A and F2 = cfw_L B be foliations. Using your
favor
- Math 518 Differentiable Manifolds I
- Fall 2013 Homework #5
(to be returned at the beginning of class, Friday, Oct. 4)
Solve the following exercises from the Lecture Notes:
5.5) Let : P2 R3 be the map dened by
([x, y, z ]) =
1
(yz, xz, xy ).
x2 + y 2 +
- Math 518 Differentiable Manifolds I
- Fall 2013 Homework #13
(to be returned at the beginning of class, Friday, Dec. 6)
Read Sections 15 and 16 from the Lecture Notes and solve the following
exercises:
15.5) Let : M N be a smooth map and let X X(M ) and
- Math 518 Differentiable Manifolds I
- Fall 2013 Homework #12
(to be returned at the beginning of class, Friday, Nov. 22)
Read Section 12 from the Lecture Notes and solve the following exercises:
12.2) Show that every left invariant vector eld in a Lie g
- Math 518 Differentiable Manifolds I
- Fall 2013 Midterm Sample Exam
(Solutions will be posted on the web page on Wednesday, October 9)
1. (25 Points) Prove or disprove the following statement: if M and N
are smooth manifolds with boundary then M N is al
- Math 518 - Fall 2013 Differentiable Manifolds I
Final Exam
(Tuesday, December 17, 7-10 pm)
Name:
1. (40 Points) On R4 with coordinates (x, y, z, w) consider the following
3 vector elds:
X1 = x
y
y
x
X2 = y
z
z
y
X3 = x
z .
z
x
Show that they span a 2-di
- Math 518 Differentiable Manifolds I
- Fall 2013 Midterm - October 11, 2013
(Duration of the exam: 50 minutes)
1. (25 Points) Prove or disprove the following statement: Let M Rn
be a manifold with boundary embedded in Rn . Then x M if and only if
for eve
- Math 518 Differentiable Manifolds I
- Fall 2013 Homework #11
(to be returned at the beginning of class, Friday, Nov. 15)
Read Section 11 from the Lecture Notes and solve the following exercises:
11.1) Give an example of a smooth distribution D of dimens
- Math 518 Differentiable Manifolds I
- Fall 2013 Homework #1
(to be returned at the beginning of class, Friday, Sep. 6)
Solve the following exercises from the Lecture Notes:
0.2) Let f : Rd Rm be a map of class C k , k = 0, . . . , . Show that
: Rd Grap