Re: Embedding theorem and uniqueness?

From: Tom Roberts (tjroberts_at_lucent.com)
Date: 11/30/04 

Date: Tue, 30 Nov 2004 15:12:00 -0600

inquirydog wrote:
> According to the embedding theorem, any manifold can be
> represented as a surface in a higher dimensional space (ie- the
> 2-manifold with constant curvature a^2 can be represented as the
> sphere of radius a in three dimensional space).

Which "embedding theorem"? -- there are several. Note I am not expert in
this area, but have some knowledge.

This also depends on what type of embedding do you mean: topological or
isometric? A topological emebdding merely maps the topology of the
original manifold into that of the larger manifold. An isometric
embedding does that and also requires that the metric of the original
manifold be the projection of the larger manifold's metric onto the
embedded image.

For any embedding to be possible, the original manifold must have a
topology compatible with the topology of the region of the larger
manifold occupied by the image. And the dimension of the larger manifold
must be greater or equal to the dimension of the original manifold.

For isometric embeddings the requirements are far more strict....

        For example, for an arbitrary 3+1-dimension Lorentzian
        manifold to be isometrically embedded into an N+M-dimension
        flat manifold, the lowest known dimension is N=88,M=2 (!).

> My question is- is
> the surface unique (up to trivial transformations such as translation,
> rotation, and curling up in a yet higher dimensional space such as
> when one rolls up a plane to a cylinder).

        [rolling up a plane into a cylinder is a major topological
         change, and should not be included in your list -- that is
         not "trivial" at all.]

This depends on the type of embedding (see above), on the topologies of
the manifolds, and for isometric embeddings also on their metrics.

For instance: consider embedding S^2 into E^3-(one point). There are two
distinct ways to do it, one encloses the deleted point and the other
does not. With suitable metrics either could be isometric.

Of course any two isometric embeddings will be "the same" in the sense
of there being an isometry between them. And any two topological
embeddings will be "the same" in the sense of there being a topological
mapping between them. By "them" I mean a region of the larger manifold
enclosing the embedded image. So your question about uniqueness needs to
be more precise about what "unique" means -- specifically do you really
mean "unique up to isometry/topological-mapping"?. I think that is a
more precise statement of your "up to trivial transformations..." (for
instance, in E^3 both rotations and translations are isometries of the
space to itself). If so, then clearly any such embedding is "unique".
But note this also implies that embedding S^2 into E^3 and into E^100
are "the same"....

But look back 2 paragraphs -- those two embeddings of S^2 are quite
clearly different, and yet the previous paragraph showed the embedding
is unique up to isometry/topological-mapping. So it should be clear that
this entire discussion is not rigorous enough.... Here there be dragons.

        This is directly related to local vs global issues....

> For instance, is there
> another surface in three dimensional space that has constant curvature
> a^2?

As you mention curvature, I'll assume you mean an isometric embedding.

If you mean embedding into flat space E^3, then I'm pretty sure any pair
of such embeddings will be congruent, and it is therefore unique in your
sense. Buf if, for instance, one embeds into E^3-(one point), then there
are two different mappings; etc.

> Are there other solutions which have constant curvature but
> eventually have a singularity at global locations?

If you mean embedding into flat space E^3, then I don't think so. But I
am not knowledgeable enough to be sure.

Tom Roberts tjroberts@lucent.com

Relevant Pages

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