ONT Re: Manifolds Of Sensuous Impressions (MOSI's)
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| The quality 'q', so far, is an entirely subjective fact
| which the feeling carries so to speak endogenously, or
| in its pocket. If any one pleases to dignify so simple
| a fact as this by the name of knowledge, of course
| nothing can prevent him. But let us keep closer
| to the path of common usage, and reserve the name
| knowledge for the cognition of "realities", meaning
| by realities things that exist independently of the
| feeling through which their cognition occurs. If the
| content of the feeling occur nowhere in the universe
| outside of the feeling itself, and perish with the
| feeling, common usage refuses to call it a reality,
| and brands it as a subjective feature of the feeling's
| constitution, or at the most as the feeling's 'dream'.
|
| William James, 'The Meaning of Truth',
| Longmans, Green, & Co., London, 1909,
| page 5-6.
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| Suppose that X is finite dimensional of dimension n, and that Y is a submanifold of dimension m.
| Then from the definition we see that the local product structure in the neighborhood of a point
| of Y can be expressed in terms of local coordinates as follows. Each point P of Y has an open
| neighborhood U in X with local coordinates (x_1, ..., x_n) such that the points of Y in U are
| precisely those whose last n - m coordinates are 0, that is, those points having coordinates
| of type
|
| (x_1, ..., x_m, 0, ..., 0).
|
| Let f : Z -> X be a morphism, and let z be in Z. We shall say that f is
| an "immersion" at z if there exists an open neighborhood Z_1 of z in Z
| such that the restriction of f to Z_1 induces an isomorphism of Z_1
| onto a submanifold of X. We say that f is an "immersion" if it is
| an immersion at every point.
|
| Note that there exist injective immersions
| which are not isomorphisms onto submanifolds,
| as given by the following example:
| ________
| / \
| / \
| | |
| | |
| \ V
| \___________________________________________
|
| (The arrow means that the line approaches itself without touching.)
| An immersion which does give an isomorphism onto a submanifold is
| called an "embedding", and it is called a "closed embedding" if
| this submanifold is closed.
|
| Lang, DARM, pages 24-25.
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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.
|
| 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.
|
| If X, Y are two manifolds, then one can give the
| product X x Y a manifold structure in the obvious way.
| If {(U_i, q_i)} and {(V_j, r_j)} are atlases for X, Y
| respectively, then
|
| {(U_i x V_j, q_i x r_j)}
|
| is an atlas for the product, and the product of compatible
| atlases gives rise to compatible atlases, so that we do get
| a well-defined product structure.
|
| Let X, Y be two manifolds. Let f : X -> Y be a map.
| We shall say that f is a "C^p-morphism" if, given x in X,
| there exists a chart (U, q) at x and a chart (V, r) at f(x)
| such that f(U) c V, and the map
|
| r o f o q^-1 : qU -> rV
|
| is a C^p-morphism in the sense of Chapter 1, Section 3.
| One sees then immediately that this same condition holds
| for any choice of charts (U, q) at x and (V, r) at f(x)
| such that F(U) c V.
|
| It is clear that the composite of two C^p-morphisms is itself
| a C^p-morphism (because it is true for open subsets of vector
| spaces). The C^p-manifolds and C^p-morphisms form a category.
| The notion of isomorphism is therefore defined ...
|
| If f : X -> Y is a morphism, and (U, q) is a chart
| at a point x in X, while (V, r) is a chart at f(x),
| then we shall also denote by
|
| f_V,U : qU -> rV
|
| the map rfq^-1 [that is, r o f o q^-1].
|
| Lang, DARM, page 22.
|
| It is also convenient to have a local terminology.
| Let U be an open set (of a manifold or a Banach space)
| containing a point x_0. By a "local isomorphism" at x_0
| we mean an isomorphism
|
| f : U_1 -> V
|
| from some open set U_1 containing x_0 (and contained in U)
| to an open set V (in some manifold or some Banach space).
| Thus a local isomorphism is essentially a change of chart,
| locally near a given point.
|
| 2.2. Submanifolds, Immersions, Submersions
|
| Let X be a topological space, and Y a subset of X.
| We say that Y is "locally closed" in X if every point
| y in Y has an open neighborhood U in X such that Y |^| U
| is closed in U. One verifies easily that a locally closed
| subset is the intersection of an open set and a closed set.
| For instance, any open subset of X is locally closed, and
| any open interval is locally closed in the plane.
|
| Let X be a manifold (of class C^p with p >= 0). Let Y be a subset of X
| and assume that for each point y in Y there exists a chart (V, r) at y
| such that r gives an isomorphism of V with a product V_1 x V_2 where
| V_1 is open in some space E_1 and V_2 is open in some space E_2,
| and such that
|
| r(Y |^| V) = V_1 x a_2
|
| for some point a_2 in V_2 (which we could take to be 0). Then it is clear
| that Y is locally closed in X. Furthermore, the map r induces a bijection
|
| r_1 : Y |^| V -> V_1.
|
| The collection of pairs (Y |^| V, r_1) obtained in the above manner constitues
| an atlas for Y, of class C^p. The verification of this assertion, whose formal
| details we leave to the reader, depends on the following obvious fact.
|
| Lang, DARM, page 23.
|
| Lemma 2.1. Let U_1, U_2, V_1, V_2 be open subsets of Banach spaces,
|
| and g : U_1 x U_2 -> V_1 x V_2 a C^p-morphism.
|
| Let a_2 be in U_2 and b_2 be in V_2
|
| and assume that g maps U_1 x a_2 into V_1 x b_2.
|
| Then the induced map
|
| g_1 : U_1 -> V_1
|
| is also a morphism.
|
| Indeed, it is obtained as a composite map
|
| U_1 -> U_1 x U_2 -> V_1 x V_2 -> V_1,
|
| the first map being an inclusion and the third a projection.
|
| We have therefore defined a C^p-structure on Y which will be called
| a "submanifold" of X. This structure satisfies a universal mapping
| property, which characterizes it, namely:
|
| | Given any map f : Z -> X from a manifold Z into X such that
| | f(Z) is contained in Y. Let f_Y : Z -> Y be the induced map.
| | Then f is a morphism if and only if f_Y is a morphism.
|
| The proof of this assertion depends on Lemma 2.1, and is trivial.
|
| Finally, we note that the inclusion of Y into X is a morphism.
|
| If Y is also a closed subspace of X, then
| we say that it is a "closed submanifold".
|
| Lang, DARM, page 24.
|
| Serge Lang,
|'Differential & Riemannian Manifolds',
| Springer-Verlag, New York, NY, 1995.
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| 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 Coagitant Impressions
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References And Incidental Nuances (RAIN)
http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Riemann.html
http://www.philosophy.ru/library/kant/01/cr_pure_reason.html
http://ez2www.com/go.php3?site=book&go=0387943382
http://hallmathematics.com/mathematics/1433.shtml
http://hallmathematics.com/mathematics/630.shtml
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