Lecture 5.
6
More examples.
The orthogonal group. Let
O (n ) = cfw_ A Mn n (R)| A A T = I .
be the group of orthogonal transformations of Rn . We claim that the orthogonal
group is a smooth manifold. To see this consider the map
f : Mn n (R) S ym n (R)
gi
18.965 Fall 04
Homework 1
Exercise 1. Prove that the grassmanians Grk (Fn ) for F = R, C, or H are
smooth manifolds.
Exercise 2. Prove that the O(n) and U (n) are smooth manifolds. Here is
one hint. Show that if A is a skew symmetric (skew hermitian) matr
Lectures 10 and 11
12
Sards Theorem
An extremely important notion in differential topology is that that of general posi
tion or genercity. A particular map may have some horrible pathologies but often
a nearby map has much nicer properties.
For example th
Lecture 8.
8
Connections
We motivate the introduction of connections in a vector bundle as a generalization
of the usual directional derivative of functions on a manifold. Given a vector eld
X and a function f on a manifold M , its directional derivative
Lecture 6.
7
Vector bundles and the differential
Consider the Grassman manifold say Gr2 (R4 ) of two planes in R4 . Let
= cfw_(, x ) Gr2 (R4 ) R4 |x .
Let p : Gr2 (R4 ) be the natural projection. The bers of p , p 1 () are
vector spaces (in this case ove
Lecture 12.
13
Stratied Spaces
.
Denition 13.1. A stratication of a topological space X is a ltraion is a de
n
composition X = i =0 Si where each of the Si are smooth manifolds (possibily
empty) of dimension i and so that
Sk \ Sk
k 1
Si .
i =0
The closur
Lecture 7.
7.2 The tangent bundle
Let M be a smooth manifold. We will associate to M a bundle T M . We will do
this concretely but there are many ways of doing this. You should read about them
all!
We know what a tangent vector in Rn .
Denition 7.3. A tan
Lecture 9.
11
The embedding manifolds in R N
Theorem 11.1. (The Whitney Embedding Theorem, Easiest Version). Let X be a
compact n manifold. Then X admits a embedding in R N .
Proof. First we construct an embedding : X R N for some large N . Let
k
cfw_ f i
18.965 Fall 2004
Homework 3
Due Friday 10/9/04
Exercise 1. Prove Rieszs lemma. The unit ball in a Banach space is compact
if and only if the Banach space is nite dimensional.
Exercise 2. Prove that the adjoint of a compact operator is compact. Prove
that
18.965 Fall 2004
Homework 4
Due Monday 10/19/04
Exercise 1. Proof that the evaluation map
ev : C k (M, Rn ) Rn
is C 1 and that it is a submersion.
Exercise 2. Prove that there is an immersion of T 2 \ cfw_pt into R2 . Prove
that there is an immersion of T
18.965 Fall 2004
Homework 5
Due Monday 11/8/04
Exercise 1. Let M be smooth manifold embedded in RN . Show that for a
residual subset of the dual space of RN the restriction on a linear functional
to M is a Morse function.
Exercise 2. Show that every Morse
Lecture 4.
5
Inverse, and implicit function theorems.
Among the basic tools of the trade are the inverse and implicit function theorems.
We will rst state them in a coordinate dependent fashion. When we develop some
of the basic terminology we will have a
Lecture 3.
4
The derivative of a map between vector spaces
Let f : V W be a smooth map between real vector spaces.
Denition 4.1. Given x V we say that f is differentiable at x if there is a linear
map L x : V W so that for all v V we have:
f (x ) f (x )
Lecture 2.
2
Smooth maps and the notion of equivalence
Let X and Y be smooth manifolds. A continuous map f : X Y is called smooth
if for all charts (U , ) for and X and (V , ) for Y we have that the composition
f 1 : (U f 1 (V ) (V )
is smooth.
Two manif
Department of Mathematics 18.965 Fall 04 Lecture Notes Tomasz S. Mrowka Lecture 1.
1
Manifolds: denitions and examples
Loosely manifolds are topological spaces that look locally like Euclidean space. A little more precisely it is a space together with a w
18.965 Fall 2004
Homework 2
Due Monday 9/27/04
Exercise 1. Let F : R R is a C 2 map with uniformly bouned rst and
second derivatives. F induces a map
F : C 0 [0, 1] C 0 [0, 1]
by composition; F (u) is the function t F (u(t) Show that F is a C 1 map.
0
0
M