Re: Smooth submanifold

On Jun 25, 1:07 am, simeon1...@xxxxxxxxx wrote:
There is a nice comment of V.I.Arnol’d on smooth manifolds in his
article “On teaching mathematics”. The definition with charts and
atlases makes me sick.
 Here it is:

“What is a smooth manifold? In a recent American book I read that
Poincaré was not acquainted with this (introduced by himself) notion
and that the "modern" definition was only given by Veblen in the late
1920s: a manifold is a topological space which satisfies a long series
of axioms.

For what sins must students try and find their way through all these
twists and turns? Actually, in Poincaré's Analysis Situs there is an
absolutely clear definition of a smooth manifold which is much more
useful than the "abstract" one.

A smooth k-dimensional submanifold of the Euclidean space R^N is its
subset which in a neighbourhood of its every point is a graph of a
smooth mapping of R^k into R^(N-k) (where R^k and R^(N-k) are
coordinate subspaces). This is a straightforward generalization of
most common smooth curves on the plane (say, of the circle x^2 + y^2 =
1) or curves and surfaces in the three-dimensional space.
Between smooth manifolds smooth mappings are naturally defined.
Diffeomorphisms are mappings which are smooth, together with their
inverses.
An "abstract" smooth manifold is a smooth submanifold of an Euclidean
space considered up to a diffeomorphism. There are no "more abstract"
finite-dimensional smooth manifolds in the world (Whitney's theorem).
Why do we keep on tormenting students with the abstract definition?
Would it not be better to prove them the theorem about the explicit
classification of closed two-dimensional manifolds (surfaces)?

It is this wonderful theorem (which states, for example, that any
compact connected oriented surface is a sphere with a number of
handles) that gives a correct impression of what modern mathematics is
and not the super-abstract generalizations of naive submanifolds of a
Euclidean space which in fact do not give anything new and are
presented as achievements by the axiomatisators.

The theorem of classification of surfaces is a top-class mathematical
achievement, comparable with the discovery of America or X-rays. This
is a genuine discovery of mathematical natural science and it is even
difficult to say whether the fact itself is more attributable to
physics or to mathematics. In its significance for both the
applications and the development of correct Weltanschauung it by far
surpasses such "achievements" of mathematics as the proof of Fermat's
last theorem or the proof of the fact that any sufficiently large
whole number can be represented as a sum of three prime numbers.

For the sake of publicity modern mathematicians sometimes present such
sporting achievements as the last word in their science.
Understandably this not only does not contribute to the society's
appreciation of mathematics but, on the contrary, causes a healthy
distrust of the necessity of wasting energy on (rock-climbing-type)
exercises with these exotic questions needed and wanted by no one.
The theorem of classification of surfaces should have been included in
high school mathematics courses (probably, without the proof) but for
some reason is not included even in university mathematics courses
(from which in France, by the way, all the geometry has been banished
over the last few decades).“

The whole text at

 http://pauli.uni-muenster.de/~munsteg/arnold.html

Simeon

Well, charts make you sick and the prospect of dealing
objects as defined in this extract makes *me* sick.
You like potayto and I like potahto.

I honestly do not see much gain in going from local charts
to local graphs. A definition of manifolds that requires
one to find an explicit embedding of the Poincaré homolgy
sphere in some R^n in order to be able to call it a
manifold apears at the very least clumsy.

While Whitney's theorem is of course true, it
is on one hand hardly the case that all manifolds
naturally appear in a specific embedding in an euclidean
space and, on the other hand, very natural manifolds
require a huge codimension in order to be embedded
(projective planes!) and those embeddings, even when
described as local graphs, are totally unnatural.

Moreover, it is often the case that global objects are
constructed by patching local constructions, and the latter
rather seldomly show up as graphs of functions!
From the construction of Teichmuller's space to the
construction of covering spaces to the construction
of the Hilbert scheme to the construction of the foliation
tangent to a distribution to any one of innumerable other
examples, that MO shows up constantly.

The student should of course be informed of Whitney's
and other theorems, and whenever useful, shown how to
put them to use.

But pretending that such an approach, whatever the intuitive
grounding it may have, is good enough for all purposes
is at best naive.

-- m

.

Relevant Pages

  • Re: Smooth submanifold
    ... article "On teaching mathematics". ... Between smooth manifolds smooth mappings are naturally defined. ... An "abstract" smooth manifold is a smooth submanifold of an Euclidean ... From the construction of Teichmuller's space to the ...
    (sci.math)
  • Re: Smooth submanifold
    ... article "On teaching mathematics". ... absolutely clear definition of a smooth manifold which is much more ... Between smooth manifolds smooth mappings are naturally defined. ... charts make you sick and the prospect of dealing ...
    (sci.math)
  • Re: Insult Jamble
    ... I think mathematics is still a construct - pi isn't "the thing ... a sufficiently smooth manifold, ... are those of the Statistics Department or of Purdue University. ... Herman Rubin, Department of Statistics, Purdue University ...
    (soc.culture.jewish.moderated)
  • Re: The Bill Hubba INQUIRY... Part2
    ... "> No - it is a device useful in the study of manifolds and other areas ... of mathematics and physics." ... No. Tensors are a mathematical concept applicable to many areas of pure, ...
    (sci.physics.relativity)
  • Re: The Bill Hubba INQUIRY... Part2
    ... "> No - it is a device useful in the study of manifolds and other areas ... of mathematics and physics." ... No. Tensors are a mathematical concept applicable to many areas of pure, ...
    (sci.physics.relativity)
Loading