Subject: Re: query (fwd)
From: Richard Hain <hain@math.duke.edu>
Date: Fri, 7 Sep 2007 23:04:01 -0400 (EDT)

> > ---------- Forwarded message ----------
> > Date: Thu, 06 Sep 2007 17:28:23 -0400
> > From: jim stasheff <jds@math.upenn.edu>
> > To: Don Davis <dmd1@lehigh.edu>
> > Cc: JimStasheff stasheff <jds@math.upenn.edu>
> > Subject: query
> >
> > KT Chen invented his theory of smooth spaces via plots in part to do
> > `differential geometry' on a loop space. His collected works would
> > contain a lot of the pieces of such a geometry but I think other
> > applications he did not get to before his early death.
> >
> > Is there anything like a textbook or ``Differential Geometry from
> > Chen's point of view'?

Chen did write a set of lecture notes on the calculus of variations
from this point of view. I read them when I was a student, but am not
sure that I still have a copy. I will take a look. I was once told
that category theorists (who were interested in categorical approaches
to differential geometry) were interested in his work, and that there
is a topos called the "Chen topos" that has something to do with his
differentiable spaces. I know of no differential geometry texts that
take Chen's approach.

Best, Dick
____________________________________________________

Subject: Re: four postings
From: Andrew Stacey <andrew.stacey@math.ntnu.no>
Date: Mon, 10 Sep 2007 15:53:47 +0200

You will probably get lots of replies about 'diffeological spaces', and in
particular references to Patrick Iglesias' book (see his homepage at:
http://math.huji.ac.il/~piz/Site/Welcome.html).  Another approach is 
Frolicher
spaces.  I'm not sure if there is a book about Frolicher spaces, but the 
book
"A Convenient Setting for Global Analysis" (Kriegl and Michor, available 
for
download from the AMS) mentions them and can be viewed as being about 
"locally
linear Frolicher spaces".  There's quite a lot of geometry in that book.

I think that Frolicher spaces and diffeological spaces (Chen's spaces) are 
one
and the same.  Frolicher spaces are specified by giving the smooth curves 
and
smooth functionals, Chen spaces by giving the plots.  However, the smooth
curves completely determine the Frolicher structure, and a plot is 
completely
determined by its restriction to the smooth curves that factor through it.
Thus both are completely determined by declaring a family of curves to be
"smooth".

Andrew