[Arisbe] Manifolds Of Sensuous Impressions (MOSI's)

[Jon Awbrey]

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Manifolds Of Sensuous Impressions (MOSI's)

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| This paper is based upon the theory already established, that the function of
| conceptions is to reduce the manifold of sensuous impressions to unity, and that
| the validity of a conception consists in the impossibility of reducing the content
| of consciousness to unity without the introduction of it.  (CSP, CP 1.545, CE 2.49).

Let me read you a story from one of my favorite books of manifolds:

| Serge Lang,
|'Differential & Riemannian Manifolds',
| Springer-Verlag, New York, NY, 1995.

But first, a message from our medium:

¤~~~~~~~~~¤~~~NOTATIONATE~NOTEFACTION~~~¤~~~~~~~~~¤

In presenting this text I am obligated to change
many Greek characters into Latin letters, and so
by way of a slightly skewed form of compensation,
I will convert Roman numerals to Arabic decimals.
Notes from the translator (me) will be placed in
square brackets, to ease the transits to English.

Let "|_|", interfixed with extra space around it,
or else "|_|<i>", antefixed, signify the union
of two sets, or of the many sets indexed by i,
respectively.

Let "|^|", interfixed with extra space on either side,
or else "|^|<i>", antefixed, signify the intersection
of two sets, or of a family of many sets indexed by i,
respectively.

Let "o", interfixed with extra space around it,
signify functional composition, interpreted in
the sense that (f o g)(x) = f(g(x)).

Note to critics who may happen to follow the style sheet
of the APA ("American Pedantical Association").  The "we"
that you see prevailing in this mannerism of mathematical
writing is not of necessity the "we" of plural authorship,
and of necessity not the "we" of birth through royal blood,
as it was discovered years ago that there is no royal robe
to mathematics, but it is the very democratic "we" of the
participatory demonstracy, and it begins to lose its title
to that with every citizen of this res publica who demurs
from their reponsibility and their right to follow along.

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Good.  Once upon a time ...

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Chapt 2.  Manifolds

Starting with open subsets of Banach spaces [think R^n for the moment],
one can glue them together with 'C^p'-isomorphisms [bijective mappings
that are continuously differentiable up to at least as far as order p].
The result is called a manifold.  We begin by giving the formal definition.
We then make manifolds into a category, and discuss special types of morphisms.
We define the tangent space at each point, and apply the criteria following
the inverse function theorem to get a local splitting of a manifold when
the tangent space splits at a point.
 
We shall wait until the next chapter to give a manifold structure
to the union of all the tangent spaces.

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 is true 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.

It is a trivial exercise in point set topology to prove that one
can give X a topology in a unique way such that each U<i> is open,
and the q<i> are topological isomorphisms.  (Lang, DARM, 20-21).

| Serge Lang,
|'Differential And Riemannian Manifolds' (DARM),
| Springer-Verlag, New York, NY, 1995, pp. 20-21.

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To be continued, and I dare say it, to be differentiated,
up to some order as yet to be predestinately determinate.

Jon Awbrey

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