Re: 2-manifold metric spaces with many symmetries

From: David Bernier (david250_at_videotron.ca)
Date: 11/26/04 

Date: Fri, 26 Nov 2004 02:17:46 -0500

W. Dale Hall wrote:
> David Bernier wrote:
>
>> A property of euclidean 2-D space is that:
>>
>> Given two congruent triangles with distinct edge-lengths, say
>> triangle(A, B, C) and triangle(A', B', C') , then there is just one
>> isometry that maps the first triangle to the second. ( A, B, C, A',
>> B', C' are points in euclidean 2-D space.) [ Property 1 ]
>>
>> I think the same is true for a torus, viewed as a finite-height
>> cylinder with vertical axis of rotation and where the upper boundary
>> of the cylinder is glued to the lower boundary.
>>
>
> The result does not hold for the torus T^2. T^2 has all the
> translation symmetries of R^2, but rotations in T^2 (i.e.,
> isometries of T^2 that fix a point) are severely limited.
[...]

That came as a surprise to me. But now I see that viewing T^2
as a 1x1 square with appropriate gluings, the farthest one can
get from (0,0) while traveling horizontally or vertically is 1/2,
whereas traveling from (0,0) at a 45 degree angle, one can
reach (1/2, 1/2) at a distance of sqrt(2)/2 > 1/2.

>
>> For a small enough open neighborhood of a point on the torus (with
>> metric as above), this will be isometric to some open set in R^2.
>>
>> For a 2-D metric space with Property 1, I'm wondering what geometries
>> are possible locally.
>>
>
> I'm not sure what you mean by a 2-D metric space. Do you
> mean an arbitrary metric space, for which the topological
> dimension is 2, or are you referring to a 2-manifold that
> has a metric?
>
> As far as "what geometries are possible locally", what
> should that mean?

If X and Y are metric spaces, O_X and O_Y points of X and Y,
and if there is an r>0 such that the open ball of radius r
centered at O_X is isometric to the open ball of radius r
centered at O_Y via an isometry f: B(r, O_X) -> B(r, O_Y)
such that f(O_X) = O_Y, then I would say that
X (near O_X) has the same geometry as Y (near O_Y).

>> I wish to exclude from consideration surfaces such as the surface of
>> a cube, but I presently don't know how to say it precisely.
>>
>
> Presumably, you want the surface to be a smooth submanifold
> of R^3.

I want the surface to be a smooth submanifold of R^n,
for some positive n.

( I had a look at "Riemannian manifold"
   here:
   http://en.wikipedia.org/wiki/Riemannian_manifold )

David Bernier

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