[Arisbe] Re: Manifolds Of Sensuous Impressions (MOSI's)
Jon Awbrey
arisbe@stderr.org
Tue, 07 Aug 2001 14:14:01 -0400
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Once the preambulatory props and supports have been set out in such a way
as to establish our subject on the appropriate numbers and aerity of feet,
my aim for the creature's tentaclive course is to chart a beeline for the
tripod or trivet where our subject might well have been able to read that
destiny from the outset, if our subject had but taken the trouble to read.
It is time to introduce the concept of "coordinates":
| If E = R^n for some fixed n, then we say that the manifold is "n-dimensional".
| In this case, a chart
|
| q : U -> R^n
|
| is given by n coordinate functions q_1, ..., q_n. If 'P' denotes a point of U,
| these functions are often written
|
| x_1(P), ..., x_n(P),
|
| or simply x_1, ..., x_n. They are called "local coordinates" on the manifold.
|
| If the integer p (which may also be infinity) is fixed throughout a discussion,
| we also say that X is a manifold.
|
| Lang, DARM, page 21.
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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.
|
| 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.
|
| 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.
|
| Serge Lang,
|'Differential & Riemannian Manifolds',
| Springer-Verlag, New York, NY, 1995.
| 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 Concreated Impressions
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