MAT 531 (Spring 1996) Topology/Geometry II

Weeks 1,2,3

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.

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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