🚀 go-pugleaf

In article <1992Nov10.000228.14551@samba.oit.unc.edu> Bruce Scott writes:
>asimov@wk223.nas.nasa.gov (Daniel A. Asimov) writes:
>
>>As has been mentioned, it *is* a theorem that all manifolds do occur
>>as subspaces of some Euclidean space: in fact, any n-manifold embeds in R^(2n).
>>
>>This is a pleasant theorem, but it would be inappropriate to take this as a 
>>definition.
>
>What about the one-dimensional helix embedded in R^3. Is the helix not a
	[...]
>Someone educate me on this.
>
>Dr Bruce Scott
-----------------------------------------------------------------------------------

A manifold is purely a topological or differential-topological notion, without
any specific metric assigned.  Your helix is the same as R^1 as a smooth manifold,
so as such it embeds not only in R^2 but even in R^1.

There is another notion, that of a Riemannian manifold.  This is a smooth manifold
supplied with a way of defining the speed of any curve at any point--as a consequence
you can define lengths of curves and distances between any two points in an appropriate 
way.  However, as a Riemannian manifold, your helix is isometric to R^1 and so embeds
isometrically in R^1 (!)  

Yet a third notion is a manifold which happens to be a metric space which arose through
embedding the manifold in Euclidean space, by measuring the distance across the Euclidean
space.  Interestingly enough, this is not very often considered in math research.  

But as *such*, with the extrinsic metric (the one obtained from the distance across R^3), 
your helix will, of course, *not* embed "isometrically" in R^2, which I believe is your 
point.  That is, this extrinsic metric on a copy of R^1 cannot be reproduced with the
extrinsic metric from some embedding of R^1 into R^2.

It would be interesting to know more about what kind of extrinsic metrics can arise 
from smoothly embedding manifolds in Euclidean space.


Dan Asimov
Mail Stop T045-1
NASA Ames Research Center
Moffett Field, CA 94035-1000

asimov@nas.nasa.gov
(415) 604-4799