Can a Möbius Strip be embedded in R3 with zero Gaussian curvature?

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Discussion Overview

The discussion centers on the possibility of embedding a Möbius Strip in R3 with zero Gaussian curvature. Participants explore the mathematical and geometric implications of such an embedding, including the nature of curvature and the properties of related surfaces like the Klein Bottle.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Ogai proposes the existence of a Möbius Strip with zero Gaussian curvature and requests parametric equations for such an embedding in R3.
  • Another participant describes a standard embedding method involving revolving a line around a circle, suggesting that this method does not yield zero curvature.
  • Ogai challenges the previous embedding by stating that its curvature is not zero, referencing the Riemann tensor as evidence.
  • One participant recalls that gluing edges of a Möbius Strip results in a Klein Bottle, which is not embeddable in R3, raising questions about the geometry of R3.
  • Another participant confirms that a Möbius Strip has only one edge and discusses the implications of embedding it and related surfaces in R3 versus R4 or S3.
  • A participant mentions a paper that may provide answers to Ogai's query about the embedding.
  • One participant asserts that a closed connected compact surface in R3 cannot be a Möbius Strip due to its properties.
  • Ogai shares a link to a thesis that discusses "Locally Euclidean Möbius Strips" in R3, suggesting it may be relevant to the discussion.

Areas of Agreement / Disagreement

Participants express differing views on the feasibility of embedding a Möbius Strip with zero Gaussian curvature in R3. There is no consensus on the existence of such an embedding, and multiple competing perspectives are presented.

Contextual Notes

Participants reference various mathematical concepts and properties related to curvature and embeddings, but the discussion remains unresolved regarding the specific equations and conditions for a zero curvature Möbius Strip in R3.

Ogai
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I would welcome the parametric equations for an embedding in R3 of a locally Euclidean Möbius Strip without self intersections nor singularities and of Gaussian curvature equal to zero. That it exists in R3 is trivial to prove: just get a strip of paper of appropriate length and width, twist and paste and you are done. Paper cannot be stretched so the intrinsic curvature of the animal is zero. You may perhaps appreciate looking at the ondulation of the Möbius Band while embedding in ordinary space. That one is the one I want to capture. :cool:

Regards,

Ogai
 
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The standard embedding is to take a line, your first coordinate, and revolve it around a circle while rotating it by \pi radians. The circle is your second coordinate, and the rotation is a function of your position on the circle. You can model this in POV-Ray using while loops, or derive the explicit parametric equation. The parametric equations for the above using a circle of radius 2 and a line (-1,1) are (x,y,z) = (2cos(t) + s*cos(t/2)*cos(t), 2sin(t) + s*cos(t/2)*sin(t), s*sin(t/2)).
 
Yes, your Möbius Strip is a sound one. Unfortunately it doesn't qualify, because its curvature is not zero. The sole component of the Riemann tensor for your example is
R1212 = 16 / [s^2+4*(2 + s*cos(t/2))^2]
which is not even constant and much less zero. The nature of the problem
was to find a "Locally Euclidean" MS.

Ogai
hypermorphism said:
The standard embedding is to take a line, your first coordinate, and revolve it around a circle while rotating it by \pi radians. The circle is your second coordinate, and the rotation is a function of your position on the circle. You can model this in POV-Ray using while loops, or derive the explicit parametric equation. The parametric equations for the above using a circle of radius 2 and a line (-1,1) are (x,y,z) = (2cos(t) + s*cos(t/2)*cos(t), 2sin(t) + s*cos(t/2)*sin(t), s*sin(t/2)).
 
I seem to recall that glueing the edges of a Möbius Strip gives a Klein Bottle, but that such is not embeddable in R3. Correct me if need be: I am a tourist here. So, does the geometry of R3 make this so? or might a Klien Bottle fit nicely in some non-Euclidian rendition of R3?
 
Please notice that a Möbius strip has only ONE single edge and not two. It is true that if you get two Mobius strips one twisted clockwise and the other anti-clockwise and you glue them along their borders you get a Klein bottle but that cannot be done in R3 without self-intersection. Yes, the geometry of R3 doesn't allow it. The same thing with a flat torus. You cannot embedd it in R3, while it is trivially embedded in R4 or, what is more, in S3, the 3-dimensional sphere.

What is enervating with the problem I proposed (to embedd a flat MS in R3) is the fact that it can be done trivially but it is so hard to write down its eqs) :cool:


benorin said:
I seem to recall that glueing the edges of a Möbius Strip gives a Klein Bottle, but that such is not embeddable in R3. Correct me if need be: I am a tourist here. So, does the geometry of R3 make this so? or might a Klien Bottle fit nicely in some non-Euclidian rendition of R3?
 
Hi Odai,
You will find your answer in this paper.
 
a closed connected compact surface in R^3 has an inside and an outside, so cannot be a mobius strip.
 
Looks like I will have 'my' Möbius Strip as soon as I get the paper worked out. Thank you so much Hyper.

hypermorphism said:
Hi Odai,
You will find your answer in this paper.