**Week 1**

Quotient Topology (Bredon I.13)

**Week 2**

Implicit Function Theorem (Bredon II.1)

**Week 3**

1. Inverse Function Theorem. Definition of smooth
(*C*-infinity) *n*-dimensional manifold
(using Bredon's Def. II.2.1). We will make use of the
note which states that any atlas can be *uniquely*
completed to a maximal atlas, to define a manifold
by giving an atlas without worrying about maximality.

*Examples*

A. R^*n* itself. (Note that maximality of the
altas means that all diffeomorphisms : *U* --> R^*n*,
(*U* open in R^*n*) would be included).

B. *S*^2. Take the sphere {*x*^2+*y*^2+*z*^2 = 1}
in R^3, let *N* and *S* be the North and South poles
(0,0,1) and (0,0,-1), and consider the atlas made up of
*U*_*N* = *S*^2 - {*N*} and
*U*_*S* = *S*^2 - {*S*}, with
*h*_*N*: *U*_*N* --> R^2 defined by
stereographic projection:
for *x* in *U*_*N*, the ray *Nx*
intersects the (*z*=0)-plane at *h*_*N*(*x*);
and *h*_*S*: *U*_*S* --> R^2
similarly defined. The change-of-coordinates
map *h*_*N* o (*h*_*S*)^(-1),
mapping R^2 - {0} to itself, is given
in polar coordinates by (*r*,theta) --> (1/*r*,theta)
(this involves
some elementary geometry) which is clearly of class
*C*-infinity. Note that if *S*^2 is oriented
by an outward pointing normal, and if R^2 is oriented as
usual, then *h*_*N* is orientation reversing. If this is corrected
by composition with the reflection (*r*,theta) --> (*r*, -theta)
then the change-of-coordinates becomes (*r*,theta) -->
(1/*r*, -theta) which is *z* --> 1/*z* in polar coordinates.

C. Projective spaces are manifolds which come with a
canonical atlas. Real projective *n*-space, the space of lines
through 0 in R^(*n*+1), admits *homogeneous coordinates*:
the line through a non-zero point (*x*_0,...,*x*_*n*)
is represented by
[*x*_0:...:*x*_*n*] where it is understood that
[*x*_0:...:*x*_*n*] and
[*a**x*_0:...:*a**x*_*n*] (*a* not 0)
are equal. The canonical atlas
is defined by *U*_*i* = {*x*_*i* not 0}
with *h*_*i*: *U*_*i* --> R^*n*
given by *h*_*i*([*x*_0:...:*x*_*n*]) =
(*x*_0/*x*_*i*,...,*x*_(*i*-1)/*x*_*i*,
*x*_(*i*+1)/*x*_*i*,
...,*x*_*n*/*x*_*i*).
The inverse is then (*h*_*i*)^(-1)(*y*_1,...,*y*_*n*)
=
[*y*_1:...:*y*_(*i*-1):1:*y*_(*i*+1):
...:*y*_*n*]. So the change of coordinate
map *h*_*j* o (*h*_*i*)^(-1) takes (*y*_1
,...,*y*_*n*) to
(*y*_1/*y*_*j*,...,1/*y*_*j*
,...,*y*_*n*/*y*_*j*),
leaving out *y*_*j*/*y*_*j*, which
is clearly of class *C*-infinity.

Exactly the same definitions and calculations work for the
complex projective spaces (note that the "*a* not 0" is now
a nonzero complex number). In particular for C*P*^1 the
canonical coordinate functions are [*z*_0,*z*_1] -->
*z*_1/*z*_0
and [*z*_0,*z*_1] --> *z*_0/*z*_1;
they are related on the overlap
by the analytic map *z* --> 1/*z*. This shows that C*P*^1 is the
same manifold as *S*^2.

2.
Let *M* be a smooth *n*-manifold with atlas
{*h*_*a* : *U*_*a* -->
R^*n*, and *N* a smooth *p*-manifold with atlas
{*k*_*b* : *V*_*b*
--> R^*p*}. A map *f*: *M* --> *N*
is said to be *smooth*
if for every applicable pair (*a*,*b*) the composition
*k*_*b* o *f* o (*h*_*a*)^(-1)
is a C-infinity map from *h*_*a*(*U*_*a* intersect
*f*^(-1)*V*_*b*) to R^*p*.
Special cases: a smooth real-valued
function on *M* and a smooth curve *c*: R --> *M*.

A *tangent vector* at a point *x* in *M* is an equivalence
class of velocity "vectors" of smooth curves passing through *x*.
Suppose *b* and *c* are both smooth curves in *M*
with *b*(0) = *c*(0) = *x*.
then *b*'(0) and *c*'(0) are equivalent if for every smooth function
*f* defined near *x*, the derivatives
(*f* o *b*)'(0) and (*f* o *c*)'(0)
are equal. The equivalence class *v* acts on smooth functions defined
near *x* by *v*.*f* = (*f* o *c*)'(0)
for a representative curve *c*.

If *x* lies in the coordinate chart *h*_*a* :
*U*_*a* --> R^*n*, then there
are *n* special tangent vectors at *x* which correspond to
partial derivatives with respect to the coordinates. More
specifically, suppose R^*n* has coordinates
(*x*_1,...*x*_*n*), and
suppose for simplicity that *h*_*a*(*x*) = (0,...0). Define
D/D*x*^*a*_i to be the velocity vector at *t* = 0 of the curve
*c*_*i*(*t*) = (*h*_*a*)^(-1)(0,...,*t*,...,0),
*t* in the *i*-th position.
These vectors form a basis for the space of tangent vectors
to *M* at *x*, because if *c* is any smooth curve with
*c*(0) = *x*,
we may write (*f* o *c*)(*t*) =
[*f* o (*h*_*a*)^(-1) o *h*_*a* o *c*](*t*)
and apply the chain rule in R^*n* to the composition of
*f* o (*h*_*a*)^(-1) with *h*_*a* o *c*.
This yields

(*f* o *c*)'(0) = \sum_*i*
[D(*f* o *h*_*a*)^(-1)/D*x*_*i*]
[(*h*_*a* o *c*)'_*i*(0)].

Now examining the definition of partial derivative shows
that [D(*f* o *h*_*a*)^(-1)/D*x*_*i*]
is exactly D*f*/D*x*^*a*_*i*; the
*n* numbers (*h*_*a* o *c*)'_*i*(0)
give the components of
*v* = (*f* o *c*)'(0) with respect to the basis
D/D*x*^*a*_1 ... D/D*x*^*a*_*n*.

If we look at the same v in the coordinate chart
*h*_*b*: *U*_*b* --> R^*n*,
it will have components (*h*_*b* o *c*)'_*i*(0)
in the new basis. These
are related to the *h*_*a* components by writing

(*h*_*b* o* *c)'_i(0) =
[*h*_*b* o (*h*_*a*)^(-1) o
*h*_*a* o *c*]'_*i*(0)

and
applying the chain rule in R^*n* again to yield

(*h*_*b* o *c*)'_*i*(0) =
\sum_*j* [D(*h*_*b* o (*h*_*a*)^(-1))_*i*
/D*x*_*j*][(*h*_*a* o *c*)'_*j*(0)].

The new components are related to the old by the matrix of
partial derivatives of the change-of-coordinates map.

This procedure exhibits the set *TM* of all tangent vectors to all
points of *M* as a smooth manifold itself. Let pi : *TM* --> *M*
associate to each tangent vector its basepoint. The calculation
shows in fact that the
open sets {pi^(-1) (*U*_*a*)} together with the homeomorphisms:
pi^(-1) (*U*_*a*) --> R^*n* x R^*n* taking a tangent vector
*v* to
*h*_*a* o pi (*v*),
*v*^*a*_1,...,*v*^*a*_*n*
(where *v*^*a*_*i* are the components
of *v* with respect to the basis D/D*x*^*a*_1 ...
D/D*x*^*a*_*n* of the
tangent space at pi(*v*)) form a smooth 2*n*-dimensional atlas.

The coordinate changes in this atlas are of a special type,
in that they respect the vector addition and scalar
multiplication in the tangent space at each point. A space
with this kind of structure is called a vector bundle;
the *tangent bundle* *TM* is special in that the base is a
manifold, the fiber dimension is the same as the dimension of
the base, and the fiber coordinate change matrix is the
jacobian of the coordinate-change map on the base. What is left?

A *k*-dimensional *coordinate vector bundle* consists of the
following objects:

pi : *E* --> *B* a continuous, surjective map from the "total space"
*E* to the "base space" *B*;

{*U*_*a*} an open covering of *B*;

*H*_*a* : pi^(-1) (*U*_*a*) -->
*U*_*a* x R^*k* a homeomorphism mapping each
pi^(-1) (*x*) to {*x*} x R^*k*;

a continuous *g*_*ab*: *U*_*a*
intersect *U*_*b* --> GL(R,*k*) for each non-empty
intersection, satisfying

*H*_*a* o (*H*_*b*)^(-1) (*x*,*v*) = (*x*,
*g*_*ab*(*x*)*v*)

for every *a*,*b*, and (*x*,*v*) in
*H*_*b*(pi^(-1)(*U*_*a* intersect *U*_*b*)).

*Examples* 0. the projection *B* x R^*k* --> *B* is a coordinate
bundle, with the one-element cover *U*_0 = *B*, *H*_0 :
*B* x R^*k* --> *B* x R^*k*
the identity map and *g*_00 = the identity matrix. This is the
*trivial* *k*-dimensional coordinate vector bundle over *B*.
Analogously, the {*U*_*a*} in general are called "trivializing
neighborhoods" and *H*_*a* is called a "local trivialization."

1. The tangent bundle pi : *TM* --> *M* of an *n*-dimensional
smooth manifold with atlas {*U*_*a*} is clearly an
*n*-dimensional coordinate vector bundle. Here
*g*_*ab*(x) = [D(*h*_*a* o (*h*_*b*)^(-1)_*i*/
D*x*_*j*](*h*_*b*(*x*)). When as here
the base is a manifold and the *g*_*ab* are smooth maps into the
general linear group, we will speak of a "smooth bundle."

Two coordinate vector bundles over the same base are
*equivalent* if the two sets of local trivializations are
compatible; an equivalence class is a *vector bundle*.

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