SUO: Re: Manifolds Of Sensuous Impressions (MOSI's)

[Jon Awbrey <jawbrey@oakland.edu>]

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SUO Work Groupers,

I will make some effort to continue the story
with no further unnecessary interrupretations.

Notation:

Let "|_|", infixed 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 "|^|", infixed with extra space around it,
or else "|^|<i>", antefixed, sign the intersection
of two sets, or of the many sets indexed by i,
respectively.

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

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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.  ...  (DARM, pages 20-21).

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

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More, Later,

Jon Awbrey

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