[Discussion] [VERY Off-Topic!: Ehresmann Connections]

[Thomas Clayton]

Dear Prof. Parker:

I _have_ been able to find my notes and ref.s - from Aug. 2003 and Dec. 2005.

I had originally seen ref. to the article in
Allendoerfer, C. B.; _Calculus of Several Variables and Differentiable Manifolds_; Macmillan Publishing Company, Inc. (1974, C.B.A.) p. 147

Newns, W. F. and Walker, A. G.; "Tangent Planes to a Differential Manifold", _Journal of the London Mathematical Society_; [31] (1956) pp.400-407


They define (I think!) a class, A, of Allowable functions in the neighborhood of a point N(P) as those whose composition with other members are within that same class.
...
In the last paragraphs, there are some notes of disagreement of others' def'n.s of tangent vectors - notably S. S. Chern.

Sincerely,

Thomas Clayton

This letter started at 3PM and was immediately interupted by a phone call. At 11+ PM, I restarted composing it and, once again, was interrupted by a text msg.(s) from the same caller! I finished it BEFORE looking at any reply from you.


--- On Tue, 3/2/10, Thomas Clayton <topcatdrc at yahoo.com> wrote:

> From: Thomas Clayton <topcatdrc at yahoo.com>
> Subject: [Discussion] [VERY Off-Topic!: Ehresmann Connections]
> To: "POSSI Discussion List" <discussion at lists.possi.org>
> Cc: "David Beran" <davemnfl at aol.com>
> Date: Tuesday, March 2, 2010, 5:25 AM
> Dear Prof. Parker:
> 
> I made the mistake of looking at WHAT the PAI ref. was to.
> 
> Its first reference, in turn, mentioned "Ehresmann
> Connections". I was crazy enough to download it and read -
> well, try to read - (some of) it.
> 
> For me, who just had gotten a 'finger hold' (mountain
> climbing metaphor) on _tangent bundles_ the last time I
> seriously looked at Math. (three years ago), it looks as if
> there are ?Tangent Tangent Bundles? (TTM) somehow related to
> second derivatives - because you mention SODEs. 
> 
> Now I - or rather, then I'd - thought I was getting pretty
> close to the finale (summit) of all this abstraction with
> this thought: Tangents get you the first D1,C1 level and
> connections seemed to be the D2,C2 (and higher D3,C3 etc)
> levels. [i.e. First derivative, second derivative, etc.].
> (Riemann, and such, are abstracted to connections, I
> thought.)
> 
> NOW you've got me scratching my head - and wondering (from
> my cravas(?) perch) just WHERE this climb goes to!
> 
> The K. Freeman ~History of Connections~ (ref. 24) looks
> like an interesting read to me, BTW.
> 
> 
> Lastly, and NOT related to your (joint) paper, 
> When I was last looking at Math, I had gone to see a paper
> by Newns and Walker about the requirement relaxation of
> C'lazy eight' to C1 for 'tangent' spaces. (I can, AND will,
> get you the exact (London J. Math circa 1955) ref.
> tomorrow.) This sits on the border between Analysis and
> Dif'l. Geo./Top.
> Do YOU know of any related subject papers?
> 
> Sincerely,
> 
> Thomas Clayton
> 
> BTW, I don't know whether you'll see it (our POSSI
> list-server may strip it) but I CC:ed my cousin Dave, a
> retired MN Math Prof. who often is in FL these days. Hence,
> Dave MN FL.
> 
> --- On Tue, 3/2/10, Phil Parker <phil at math.wichita.edu>
> wrote:
> 
> > From: Phil Parker <phil at math.wichita.edu>
> > Subject: Re: [Discussion] Hard Drive Partition Sizes
> > To: "POSSI Discussion List" <discussion at lists.possi.org>
> > Date: Tuesday, March 2, 2010, 1:21 AM
> 
> ... <snip> ...  <---- BTW, I got this idea
> from our own, James Cannon!
> 
> > 
> > -- 
> >    Best wishes,
> >     Phil Parker
> >
> -----------------------------------------------------------------------------
> > URL http://www.math.wichita.edu/~pparker/  
> > PAI http://arxiv.org/a/parker_p_1
> > Random quote:
> >   F u cn rd ths u cnt spl wrth a dm!
> 
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