ONT Re: Differential And Riemannian Manifolds
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DARM. Note 2
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For ease of reference, I repeat here the definition of
an "atlas" of class C^p, and then I pick up a few more
definitions from the text.
| Let X be a set. An "atlas" of class C^p (p >= 0) on X
| is a collection of pairs (U_i, q_i) (i ranging in some
| indexing set), satisfying the following conditions:
|
| AT 1. Each U_i is a subset of X and the U_i cover X.
|
| AT 2. Each q_i is a bijection of U_i onto an open subset q_i (U_i) of
some Banach space E_i, and for any i, j, [we have the fact that]
| q_i (U_i |^| U_j) is open in E_i.
|
| AT 3. The map
|
| (q_j) o (q_i)^(-1) : q_i (U_i |^| U_j) -> q_j (U_i |^| U_j)
|
| is a C^p-isomorphism for each pair of indices i, j.
|
| Lang, DARM, p. 20.
An atlas, of course, is a collection of charts:
| Each pair (U_i, q_i) will be called a "chart" of the atlas.
| If a point x of X lies in U_i, then we say that (U_i, q_i)
| is a "chart at x".
|
| Lang, DARM, p. 21.
Below is a paradigmatic picture of the manifold situation with
respect to a typical pair of charts, (U_i, q_i) and (U_j, q_j).
In this Figure and elsewhere, I will make use of the notations
U_ij = U_i |^| U_j and q_ij = (q_j) o (q_i)^(-1).
o-----------------------------------------------------------o
| X |
| |
| o-------------o o-------------o |
| / \ / \ |
| / o \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| o o o o |
| | | U_i | | |
| | | | | |
| | U_i | |^| | U_j | |
| | | | | |
| | | U_j | | |
| o o o o |
| \ \ / / |
| \ \ / / |
| \ \ / / |
| \ o / |
| \ | / \ | / |
| o------|------o o------|------o |
| | | |
| | | |
o--------------------|-----------------|--------------------o
| |
q_i | | q_j
| |
o--------------------|-----o o-----|--------------------o
| E_i v | | v E_j |
| | | |
| o----------o | | o----------o |
| / \ | | / \ |
| / o | | o \ |
| / / \ | | / \ \ |
| / / \ | | / \ \ |
| o o o | | o o o |
| | | | | q_ij| | | | |
| | | ------------------> | | |
| | | | | | | | | |
| | q_i (U_ij) ----- | | | | ----- q_j (U_ij) | |
| | | | | | | | | |
| o o o | | o o o |
| \ \ / | | \ / / |
| \ \ / | | \ / / |
| \ o | | o / |
| \ / | | \ / |
| o----------o | | o----------o |
| | | |
| | | |
o--------------------------o o--------------------------o
[Figure 1. Manifold X with Charts (U_i, q_i) and (U_j, q_j)]
We find next the need for a notion of "compatibility"
among and between different atlases and their charts:
| Suppose that we are given an open subset U of X and a topological isomorphism
| q : U -> U' onto an open subset of some Banach space E. We shall say that
| (U, q) is "compatible" with the atlas {(U_i, q_i)} if each map (q_i)o(q^-1)
| (defined on a suitable intersection as in AT 3) is a C^p-isomorphism.
|
| Two atlases are said to be "compatible" if each chart of one is compatible with
| the other atlas. One verifies immediately that the relation of compatibility
| between atlases is an equivalence relation. An equivalence class of atlases
| of class C^p on X is said to define a structure of "C^p-manifold" on X.
|
| If all the vector spaces E_i in some atlas are toplinearly isomorphic,
| then we can always find an equivalent atlas for which they are all equal,
| say to the vector space E. We then say that X is an "E-manifold" or that
| X is "modeled" on E.
|
| Lang, DARM, p. 21.
|
| Serge Lang,
|'Differential & Riemannian Manifolds',
| Springer-Verlag, New York, NY, 1995.
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