From: kramsay@aol.commangled (Keith Ramsay)
Subject: Re: What is differential geometry (esp. w.r.t computability theory)?
Date: 04 Dec 2000 02:42:44 GMT
Newsgroups: sci.math
Summary: [missing]

In article <903is5$bca$1@nnrp1.deja.com>, lemma_one@my-deja.com writes:
|I doubt some mathematician
|woke up one day and thought, "today I'm going to define a manifold as a
|space in which... and then I'm going to create a branch of mathematics
|called differential geometry where we study these objects." What gave
|rise to their study? Why was their investigation natural and inevitable?
|How old is this branch?

Well, if you find group theory "natural", I hope you have no trouble
with the distinction between "discrete" and "continuous" being
natural. Mathematicians appear to have gotten by with talking more
vaguely of "continuous spaces" for a long time before they got down to
defining the various specific types.

Within the continuous spaces, the key subclass here is the kind of
space on which you can put coordinates. That's essentially what is
meant by "looking like a piece of R^n": you can parameterize that
little piece with n parameters. We say "little" when often that could
be replaced by "almost the whole manifold" because all we really need
to begin with is to ensure that the point is fully enclosed by the
coordinates. If I pick a point on a surface and ask for coordinates
"around" that point, giving me coordinates on a curve through the
point hardly will be satisfactory. We also can't afford to ask for
coordinates on the whole manifold, because there are typically
features of the manifold which prevent that. For instance, on a sphere
no one coordinate system quite serves, because any coordinate system
covering the whole sphere degenerates somewhere. Two coordinate
systems, though, one for all but the south pole and one for all but
the north pole, works fine.

A different set of issues arises when you work with spaces that are
somewhat like manifolds, but are allowed to have singularities, like
corners, self-intersections, conical points and so on, and it seems
useful to handle the two sorts of issues separately, to some extent.
There are generalizations of manifolds which may have these other
features, but they tend to require more work to deal with (and knowing
facts about the smooth pieces of the more general space helps).

Then there's the issue of what kind of functions are involved. To make
useful sense, the different coordinates you apply to your space have
to have some kind of mutual compatibility between them. This is done
by talking about "transition functions" between them, the functions
which convert from one coordinate system to another. Depending upon
what kind of functions you allow, you get various specific types of
manifolds: continuous, smooth, analytic, or complex analytic, and so
on. In some languages, the same word is used for "manifold" as for
"algebraic variety", but with a different adjective, and I think that
makes good sense; it's the analog of "manifold" relevant for algebraic
functions.

The distinctions between those special variations on the idea of
"manifold" are a little technical. But they're natural enough because
the distinctions between the types of function are natural too.
"Differential geometry" is called that because of the importance of
being able to take derivatives of the transition functions. "Algebraic
geometry" is called that because of the importance of the transition
functions being algebraic.

I think the geometers would be amused to hear that their subject was
being called a branch of analysis! Differential geometry is certainly
closely related to analysis (because of all the differentiable
functions), but I would still call it a branch of geometry.

I don't know why you suspect it of being related to computability
theory. Certainly we can consider computability questions in it, like
in any other field of mathematics, but I think most mathematicians
would consider these to be two of the least closely related fields
in mathematics.

Keith Ramsay