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
[ax_0:...:ax_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 CP^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 CP^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/Dx^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)/Dx_i]
[(h_a o c)'_i(0)].
Now examining the definition of partial derivative shows
that [D(f o h_a)^(-1)/Dx_i]
is exactly Df/Dx^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/Dx^a_1 ... D/Dx^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
/Dx_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/Dx^a_1 ...
D/Dx^a_n of the
tangent space at pi(v)) form a smooth 2n-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/
Dx_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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