Re: Topologically what is a torus in 2-d; difference between torus and sphe
- From: a_plutonium@xxxxxxxxxxx
- Date: 6 Dec 2005 00:06:19 -0800
Martin Wanvik wrote:
> I do not have this clear in my mind as yet. I am not
> good with
> topology.
> I reckon a torus is a Riemannian geom object. Is that
> true?
Yes, in the sense that a torus is a smooth manifold and any smooth
manifold admits a Riemannian metric.
> That a
> torus can be a model for Riem geom axioms.
Not sure what to make of this....
> A sphere is a model of Riem geom and I guess a sphere
> with a puncture
> hole is also a model.
> But topologically a sphere with a puncture hole is
> altogether different
> from a torus. Why should that be?
A punctured 2-sphere is homeomorphic ("topologically equal") to a
2-disk, which is not homeomorphic to a torus.
Proving it seems more difficult, and may require algebraic topology.
(They have non-isomorphic homology groups, hence they can't be
homeomorphic)
> A sphere with a puncture hole would be like a letter
> C in 2-dimensions.
> Is that true?
Yes, a punctured 1-sphere imbedded into R^2 may look like this. It
would also be a simple open interval in R.
> What would a torus in 2-dimensions thus be? Would it
> be like the C only
> a hole in the other side where you end up with two
> curves such as ( and
> ) only turned around 90 degrees.
A torus is by definition a 2 dimensional manifold. If you mean how it
would look if you tried to map it into the plane? In short, this would
depend on the mapping, but there is no way to picture a 2-torus
accurately in the plane, it requires 3 dimensions.
A.P. writes: Thanks for those answers. Thinking about it today I felt
sure that the torus is the best model for Lobachevsky geometry even
though as you say it admits a Riemannian metric.
Martin, can you help me out in showing that a torus satisfies all of
the axioms of Lobachevsky geom and thus is a perfect model for that
geometry?
What would the lines be?
We already have the disc in the plane as a model of Lobachevsky and so
the torus is just that same model only 3rd dimensional rather than 2
dimensional.
I need that result because I want to eventually get to the statement
that
Riem Model + Loba Model = Eucl 3rd dimension
This would be intuitively seen as the sphere filling in the hole of the
torus and thus completing 3rd dimensional Euclidean geom.
Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
.
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