MATH 213B, SPRING 2011
HOMEWORK 3
1. Let X, Y be smooth vector elds on a smooth manifold M and f, g : M R smooth
functions. Show that
[f X, gY ] = f g [X, Y ] + f (Xg )Y g (Y f )X.
2. Let (U, ) be a chart for M and denote the corresponding local frame for
WHEN ARE VECTOR FIELDS COORDINATE VECTOR FIELDS?
SLOBODAN N. SIMIC
Let M be a smooth manifold of dimension n and X1 , . . . , Xn smooth vector elds on M .
Question. Under what conditions are there local coordinates such that each Xi is the coordinate
vect
THREE FACES OF THE LIE BRACKET
SLOBODAN N. SIMIC
Intuitively speaking, the Lie bracket [X, Y ] of smooth vector elds X, Y on a smooth manifold
M measures the rate of change of the vector eld Y along X. There are many ways to think of
[X, Y ]. Here we desc
MATH 213B, SPRING 2011
HOMEWORK 3 HINTS
4. The push-forward of a vector eld X on M by a dieomorphism F : M N is the vector
eld F (X) on N dened by
F (X)q = Tp F (Xp ),
for any q N , where p = F 1 (q). Show that if X, Y are smooth vector elds on M , then
F
MATH 213B, SPRING 2011
HOMEWORK 5 HINTS
2. Let (M, g) be a Riemannian manifold. If f : M R is a smooth function such that
|grad f | = 1, show that the integral curves of the gradient vector eld grad f are geodesics.
Hint: Let : [0, a] M be a gradient ow l
MATH 213B, SPRING 2011
HOMEWORK 1
1. Let f : X Y be a continuous bijection between topological spaces X, Y . If X is compact
and Y Hausdor, show that f is a homeomorphism. Show by example that the conclusion
fails if X is not compact.
2. Show that every r
MATH 213B, SPRING 2011
HOMEWORK 4
1. Let (M, g) be an oriented Riemannian manifold of dimension n.
(a) Show that there is a unique exterior n-form on M such that p (E1 , . . . , En ) = 1, for
all p M , whenever (E1 , . . . , En ) is an orthonormal basis f
MATH 213B, SPRING 2011
HOMEWORK 5
1. Let be a linear connection on a smooth manifold M . Dene a map : T (M ) T (M )
T (M ) (recall that T (M ) is the space of smooth vector elds on M ) by
(X, Y ) =
XY
YX
[X, Y ].
(a) Show that is a 2 -tensor eld, calle
AN INTRINSIC PROOF OF THE GAUSS-BONNET THEOREM
SLOBODAN N. SIMIC
The goal of these notes is to give an intrinsic proof of the Gau-Bonnet Theorem, which asserts
that the total Gaussian curvature of a compact oriented 2-dimensional Riemannian manifold is
in
MATH 213B, SPRING 2011
HOMEWORK 2
k
1. Let Mmn (R) be the space of all real m n matrices and Mmn (R) the subset of all of those
m n matrices whose rank is k. Note that Mmn (R) can be identied with Rmn and is
thus a smooth manifold.
k
Show that Mmn is an o
WHAT IS. A DIFFERENTIAL FORM?
SLOBODAN N. SIMIC
1. Why differential forms?
Why are dierential forms useful? Here are a few reasons:
They provide a unied language for treating all the important results of vector calculus, such as the
generalized fundament