2-manifold metric spaces with many symmetries

From: David Bernier (david250_at_videotron.ca)
Date: 11/22/04 

Date: Mon, 22 Nov 2004 03:19:38 -0500

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.

For S^2 embedded in R^3 and where distances are measured
along arcs of great circles, I think Property 1 also holds.

For the hyperbolic plane, I don't know whether it
has Property 1.

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 wish to exclude from consideration surfaces such as
the surface of a cube, but I presently don't know
how to say it precisely.

David Bernier

Relevant Pages

  • Re: 2-manifold metric spaces with many symmetries
    ... >> isometry that maps the first triangle to the second. ... > I'm not sure what you mean by a 2-D metric space. ... and if there is an r>0 such that the open ball of radius r ... David Bernier ...
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  • Re: -- The "most natural" completion of a metric space
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  • Re: isometries, and symmetry groups
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