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HW 3 SOLUTIONS, MA518
1. Problem 1
(1) First, we check that the map
F : t (f (t), g(t) =
et + et et et
,
2
2
is injective. For distinct t1 6= t2 , it is necessary that t1 = t2 for f (t1 ) = f (t2 ) to
hold. But then, g(t1 ) = g(t1 ) and so the map is inje
HW 5 SOLUTIONS, MA518
1. Problem 1
The sphere S 2n1 is the set of vectors in Cn with unit norm i.e. vectors z = (z1 , , zn )
such that |z|2 =| z1 |2 + + | zn |2 . Consider the parameterized family of maps ft :
S 2n1 S 2n1 defined by
ft (z) = eit z
Then ft
HW 7 SOLUTIONS, MA518
Problem 1
(a): The Lie derivative of is the differential form given by
t
t0
t
LX = lim
For vector fields Y1 , , Yk this means
(Dt (Y1 ), , Dt (Yk ) (Y1 , , Yk )
t0
t
Now we have the following computation: First,
LX (Y1 , , Yk ) = li
Math 518 Differentiable Manifolds I
Assignment 5, Due Thursday November 18
1. Let K and L be embedded submanifolds of a manifold M and suppose
that their intersection KL is also an embedded submanifold of M . Then
K and L are said to have clean intersecti
Math 518 Differential Manifolds I
Assignment 5, Due Thursday October 22
1. Show that the antipodal map x x of S n is homotopic to the identity
if n is odd. Hint: Start with n = 1 by using the linear maps
cos t sin t
.
sin t cos t
2. Construct a nonvanishi
Math 518 Differentiable Manifolds I
Assignment 4, Due Thursday October 21
1. Let X be a smooth vector field on S 1 which vanishes at p S 1 . Choose
a chart (U, ) around p with (p) = 0. In the corresponding local coordinates X has the form
f (x) |x
x
where
Math 518 Differential Manifolds I
Assignment 4, Due Thursday October 7
1. For a submanifold L of M let i : L M be the inclusion map. Prove
that i (x) is the inclusion map of Tx L into Tx M.
2. If U is an open subset of M prove that Tx U = Tx M for all x U
HW 2 SOLUTIONS, MA518
1. Problem 1
Let p be a point in U . Since F is a submersion, by the implicit function theorem,
there are local co-ordinates (x1 , , xn ) around p (i.e. with p = (0, , 0) and local coordinates (x1 , , xm ) around F (p) (i.e. with F (
Math 518 Differentiable Manifolds I
Assignment 6, Due Thursday November 19
1. Let A be a linear map of an m-dimensional vector space V to itself, and
let m (V ). Compute A .
2. A non-zero k-form k (V ) is called decomposable if = 1 . . . k
where the j are
Math 518 Differentiable Manifolds I
Assignment 7, Due Tuesday December 8
1. (a) For a vector field X on a manifold M and a k-form on M, deduce
what the definition of LX , the Lie derivative of with respect to
X, should be.
(b) For a vector field X on M on
Math 518 Differentiable Manifolds I
Assignment 3, Due Tuesday October 12
1. Consider the 3-sphere S 3 R4 . Using the isomorphism R4
= C2 we
obtain the inclusion : S 3 C2 r cfw_0. Composing with the projection
map q : C2 r cfw_0 CP 1 we then get the map
p
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HW 6 SOLUTIONS, MA518
1. Problem 1
Since dim(V ) = m, we have dim(m (V ) = 1 with generator 0 = dx1 dxm in the
coordinates on V with respect to a basis (e1 , , em ). So it is enough to compute A 0 . Let
A = (aij ) be the matrix form. We have A dxi (ek ) =
Math 518 Differentiable Manifolds I
Some Practice Problems
The take-home midterm will be distributed on Thursday Oct.28 and is
due at the start of class (11:00AM sharp) on Tuesday Nov.2. Below are some
practice problems.
1. If F : M N is a submersion and
Math 518 Differential Manifolds I
Assignment 2, Due Tuesday Sept 15
1. If F : M N is a submersion and U M is open, show that F (U) is
open in N.
2. Let f : R R be a local diffeomorphism. Prove that the image of f is
an open interval and that f maps R diff
Math 518 Differential Manifolds I
Assignment 3, Due Thursday Sept 24
1. (a) Verify that the map
t 7
et + et et et
,
2
2
is an embedding of R into R2 .
(b) Determine if the map F : R2 R3 given by
F (x, y) = (x cos y, x sin y, x)
is an immersion.
(c) Check
- 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 #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