ONT Re: Manifolds Of Sensuous Impressions (MOSI's)
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Manifold SIG,
It will be useful to keep at the ready a compendium
of the most essential elements of our subject, and
so I will maintain what is needed in a cumulative
appendix to these notes.
I continue the story with further readings from Lang's DARM,
knitting up a few more strands of terminology into our yarn.
Recall the definition of an atlas:
An "atlas of class C^p (p >= 0)" on a set X is a collection of pairs (U_i, q_i),
satisfying the conditions AT 1, AT 2, AT 3, (vide syllabus at end of this note).
Naturally enough, however much artifice may have gone into its natural naming,
an atlas is conceived and executed all in order to collect a number 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.
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 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, page 21
E-nough For E-nonce,
Jon Awbrey
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| 2. Manifolds
|
| 2.1. Atlases, Charts, Morphisms
|
| 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, [it holds 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, page 20
| o---------------------------------------o o-------------------o
| | X | | E_i |
| | | | |
| | | | o |
| | | | / \ |
| | o | | / \ |
| | / \ | | / \ |
| | / \ | | / \ |
| | / \ q_i | | / q_i U_i \ |
| | / o---------------------->| o o o |
| | / \ | | \ / \ / |
| | / \ | | \ / \ / |
| | / U_i \ | | o o |
| | / \ | | \ / |
| | / \ | | \ / |
| | o o o | | o |
| | \ / \ / | | |
| | \ / \ / | | |
| | \ / U_i \ / | o---------|---------o
| | \ / \ / | |
| | o |^| o | q_j o q_i^-1
| | / \ / \ | |
| | / \ U_j / \ | o---------v---------o
| | / \ / \ | | E_j |
| | / \ / \ | | |
| | o o o | | o |
| | \ / | | / \ |
| | \ / | | / \ |
| | \ U_j / | | o o |
| | \ / | | / \ / \ |
| | \ / | | / \ / \ |
| | \ o---------------------->| o o o |
| | \ / q_j | | \ q_j U_j / |
| | \ / | | \ / |
| | \ / | | \ / |
| | o | | \ / |
| | | | \ / |
| | | | o |
| | | | |
| | | | |
| o---------------------------------------o o-------------------o
|
| Figure 1. Manifold Of Coordinated Impressions
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