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Does the concept of a Möbius torus make sense: taking a cylinder (instead of a rectangle as in the case of the Möbius strip) and twisting it before joining its ends? Or will the resulting twisted torus be indistinguishable from the normal torus in any relevant respect?

[This equivalent to the well-known Möbius strip should be called Möbius cylinder but it would have so much in common with a torus that I preferred to call it a Möbius torus.]

Embedded in Euclidean space the twisted and untwisted torus "look" the same - opposed to Möbius strip and cylinder -, the difference would be only in their intrinsic properties. But can there be such differences? And how do I specify them?

PS: I posted a follow-up question here.

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This sounds suspiciously like the Klein bottle. –  Fredrik Meyer Oct 20 '12 at 17:49
    
Are you looking for the Klein Bottle, which joins the ends of a cylinder in the opposite orientation to a torus? –  Mark Bennet Oct 20 '12 at 17:51
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A mere twist before identifying the ends of the "hose" again produces an ordinary torus with a locally euclidean metric, but its global conformal type has changed. If you identify the ends reversing the orientation you get a Klein bottle, which is a nonorientable surface. –  Christian Blatter Oct 20 '12 at 18:21
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Just to add to Christian's comment, a twist will be isotopic to the identity map, and so the resulting space will be homeomorphic to a regular torus. Think about it: if you can do it in R^3, it's not a Klein bottle. You need a degree -1 map on the circle boundaries to get the Klein bottle. –  user641 Oct 21 '12 at 4:00
    
@Christian Blatter: Can you please help me to understand better: the simply twisted torus (which I tried to ask for) has a locally euclidean metric (just like the Klein bottle), is homeomorphic to the regular torus, but has another conformal type. What exactly is the conformal type? Does it make sense to think of (closed) geodesics in this context? –  Hans Stricker Oct 23 '12 at 14:14
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up vote 10 down vote accepted

As remarked above: What you get is the Klein bottle. Put differently: The result is what you get when you take two Möbius strips (which both have one boundary) and glue both boundaries together (which does not work when embedded in 3d space but works in theory). See this image from http://plus.maths.org/content/imaging-maths-inside-klein-bottle): enter image description here

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They might not meet all the technical details of the definition; but as long as you're willing to overlook the defect where it punches through its side, you can buy glass klein bottles as conversation pieces even if you're trapped in 3-space. –  Dan Neely Oct 20 '12 at 20:51
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