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Re: torus?


As any topologist knows, a doghnut is equivalent to a coffee
cup, so why get excited about ANY of this.


A thought about one of David Bowman's posts. And besides I like (need)
coffee; in order to produce some theorems :-)



I suspect that the 'ideal' torus is an 'ideal' cylinder with it's ends
joined in a particular way. Join 'em the OTHERway to get your basic
Klein Bottle.

Since the ideal cylinder isn't really curved, then the
doughnut isn't either.



I'm not sure that your conclusion is correct.

I can make a cylinder with a flat sheet of paper and scotch tape *without*
and folds or creases or rips or tears*, which is sufficient I believe for
deciding the cylinder has no intrinsic curvature.

Take you cylinder and try to curve it around to tape the ends together (even
a cloth one), my guess is that there will necessarily be folds or creases or
tears implying non-zero intrinsic curvature.