ONT Re: Manifolds Of Sensuous Impressions (MOSI's)
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| And let us also, to escape entanglement with
| difficulties about the physical or psychical
| nature of its "object", not call it a feeling
| of fragrance or of any other determinate sort,
| but limit ourselves to assuming that it is a
| feeling of 'q'.
|
| William James, 'The Meaning Of Truth',
| Longmans, Green, & Co., London, 1909,
| page 3.
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| The collection of C^p-manifolds will be denoted by 'Man^p'.
| If we look only at those modeled on spaces in a category $U$
| then we write 'Man^p($U$)'. Those modeled on a fixed E will
| be denoted by 'Man^p(E)'. We shall make these into categories
| by defining morphisms below.
|
| Let X be manifold, and U an open subset of X. Then it is possible,
| in the obvious way, to induce a manifold structure on U, by taking
| as charts the intersections
|
| (U_i |^| U, q_i | (U_i |^| U)).
|
| [Notation. "f | S" indicates the function f as restricted to the set S.]
|
| If X is a topological space, covered by open subsets V_j, and if we are
| given on each V_j a manifold structure such that for each pair j, j' the
| induced structure on V_j |^| V_j' coincides, then it is clear that we can
| give to X a unique manifold structure inducing the given ones on each V_j.
|
| Lang, DARM, pages 21-22.
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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.
|
| 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.
|
| 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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