Math 225B: Dierential Geometry, Homework 3
Ian Coley
January 22, 2014
Problem 1.
Prove that every vector bundle over Euclidean space is trivial.
Solution.
We use the concept of a pullback bundle. Suppose that F0 , F1 : N M are smoothly
homotopic maps and
Math 225B: Dierential Geometry, Homework 8
Ian Coley
February 26, 2014
Problem 11.1.
Find H k (S 1 S 1 ) by induction on the number n of factors.
Solution.
We claim that H k (T n ) = n . For the base case, we know that H 0 (S 1 ) = H 1 (S 1 ) = R, which
k
Math 225B: Dierential Geometry, Homework 7
Ian Coley
February 20, 2014
Problem 8.17.
(a) Let M n and N m be oriented manifolds, and let and be an n-form and an m-form
with compact support, on M and N , respectively. We will orient M N by agreeing
that v1
Math 225B: Dierential Geometry, Homework 6
Ian Coley
February 13, 2014
Problem 8.7.
Let be a 1-form on a manifold M . Suppose that
Show that is exact.
c
= 0 for every closed curve c in M .
Solution.
We claim that this condition is equivalent to the follo
Math 225B: Dierential Geometry, Homework 5
Ian Coley
February 6, 2014
Problem 7.8.
(a) Let 2 (V ). Show that there is a basis 1 , . . . , n of V such that
= (1 2 ) + + (2r1 2r ).
(b) Show that the r-fold wedge product is non-zero and decomposable, and th
Math 225B: Dierential Geometry, Homework 4
Ian Coley
January 30, 2014
Problem 6.3.
(a) In the proof of Proposition 2, show that
f
xi
=
p
y i
.
f (p)
(b) Complete the proof of Proposition 2 by showing that if
n
i
,
y i
i
Y =
,
xi
i=1
so that
n
X=
i=1
i
i
i
Math 225B: Dierential Geometry, Final
Ian Coley
March 15, 2014
Problem Spring 2011, 1.
Show that if X is a smooth vector eld on a (smooth) manifold of dimension n and if Xp is
nonzero for some point of p, then there is a coordinate system dened in a neigh
Math 225B: Dierential Geometry, Homework 2
Ian Coley
January 17, 2014
Problem 5.10
(a) Prove that
LX (f ) = Xf + f LX
LX [(Y )] = (LX )(Y ) + (LX Y ).
(b) Reformulate Proposition 8 with the denition
1
(LX Y )(p) = lim [(h Y )p Yp ].
h0 h
Solution.
(a) We
Math 225B: Dierential Geometry, Homework 1
Ian Coley
January 10, 2014
Problem 3.12.
(a) Let Fp be the set of all C functions f : M R with f (p) = 0, and let : Fp R be
a linear operator with (f g) = 0 for all f, g Fp . Show that has a unique extension
to a