Questions tagged [mobius-band]

The Möbius band or Möbius strip is a surface with only one side and only one boundary. The Möbius strip has the mathematical property of being non-orientable. It is named after the German mathematician August Ferdinand Möbius.

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Which space do we obtain if we take a Möbius strip and identify its boundary circle to a point?

I know that the boundary circle of a Möbius strip is actually formed by the horizontal sides of $[0 ,1] \times [0,1]$.If we identify all the points of the 1st horizontal side to a single point and do ...
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How can the Möbius band be an image of a parametrization if it is not orientable?

I'm using a book of analysis on $\mathbb{R}^n$ with the following definitions: m-dimensional parametrization of class $C^k$ of $V\subseteq \mathbb{R}^n$: a homeomorphism $\phi:V_0\rightarrow V$ of ...
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Mobius strip and homeomorphism of a circle [closed]

I'm assigned a homework problem as follows: Let $X$ be the Möbius strip, obtained as a quotient space of $[0,1] \times [0,1]$ (with subspace topology of $\mathbb{R}^2$) by identifying the pairs of ...
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Intuition behind equivalence of two identifications obtained from Möbius band.

Consider the real projective plane $\Bbb RP^2$ which can be realized as a quotient space obtained from a square by identifying points on each pair of it's opposite edges in reverse order. It has been ...
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Proving a result concerning quotient spaces.

Let $M$ be the Möbius strip and $C$ it's boundary circle. Prove that $M/C$ is homeomorphic to $\Bbb RP^2.$ I know that $\Bbb RP^2 \approx D^2/x \sim -x, x \in \partial D^2.$ So if we can able to get ...
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What is the tangent bundle of Mobius band?(It is trivial or not?)

I know that Mobius band is a quotient space of unit square with the equivalence relation $(0,t)\sim (1,1-t)$. I want to find the tangent bundle of Mobius band upto diffeomorphism. I am not getting any ...
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Is there a fixed-point free involution of the Möbius strip?

Is there a fixed-point free continuous involution of the Möbius strip? (Meaning a function $f:M\to M$ such that $f\circ f={\rm id}$ and $f(x)\ne x$ for all $x$.) The Lefschetz fixed-point theorem is ...
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Möbius strip is non-orientable from Gallot's "Riemannian Geometry"

I was working on the proof that Möbius strip is non-orientable using the following definition: A manifold $M$ is called orientable, if there exist a smooth atlas $(U_i, \phi_i)_{i \in I}$ such that $$\...
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immersion from mobius band to $\mathbb{R^3}$

find an immersion from mobius strip to $\mathbb{R^3}$ . One way to represent the Möbius strip embedded in $\mathbb{R^3}$ is by the parametrization. $$ \begin{array}{l} x(u, v)=\left(1+\frac{v}{2} \...
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Is a thick Möbius strip genus 2?

A cylinder has genus 0, but a thick cylinder has genus 1. This gets me into wondering, since a Möbius strip has genus 1, would a thick Möbius strip have genus 2? I can't quite comprehend if the ...
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The boundary of a closed Mobius band

The Mobius band is defined by the quotient: $r:I\times I\rightarrow M$, with the equivalence relation $(0,x)\sim (1,1-x)$ for all $x\in I$. The boundary of Mobius band $M$ is defined as the set of ...
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Attaching disks and contracting is equivalent to taking quotients.

Take a Mobius band $M$ and attached a disk $D^2$ along its boundary, which is a copy of $S^1$ embedded in $M$. It is known that what we obtain is something homeomorphic to the projective plane. Now ...
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Topology of Mobius Ring

I'm having some difficulty with the finer points of the construction of the topology of the Mobius ring from Tu's manifolds. Given the square $ R = \{(x,y) \in \mathbb{R}^2 | \, 0 \leq x \leq 1, -1 &...
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2-dimensional manifold is orientable if and only if it does not contain the Möbius band

Criterion for non-orientability: a manifold $M$ is non-orientable if and only if it contains a "bad" sequence of charts, i.e. $\{U_i,\varphi_i\},i=1,\ldots,n$ such that the transition maps $\...
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how to chart a moebius strip

This is a question from a tutorial for topology manifold. It is stated as: How many charts do you need to cover the Moebius strip that has a river printed on it? The chart was defined as"Chart: -...
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Proving a function between quotient spaces is continuous

This is a question related to a problem previously asked here and also here. It states that the infinite strip $\mathbb{R}\times\left[\frac{-1}{2},\frac{1}{2}\right]$ under the group action $m\cdot(x,...
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Möbius Strip + Möbius Strip = Klein Bottle, What about Klein Bottle + Klein Bottle =?

We know that 2 Möbius-Strips can be joined edge-wise to eliminate that edge producing 0 Edge topological structure. ML (Möbius Left) + MR (Möbius Right) = KOJ (Simple "inverted sock" Klein ...
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Does this count as a loop on Möbius strip

Suppose we take the Möbius strip as $X = \frac{[0, 1]\times[0, 1]}{\sim}$ with usual equivalence relation. If I define $\alpha: [0, 1] \rightarrow X$ by $x \rightarrow [(x, 1/2)]$, is this a loop? ...
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Is this image a direct sum of two Moebius bands?

If we just look at the exterior surface of this, and view it as a 2-manifold, is this the direct sum of two Mobius bands? I see that this object has 2 sides and is orientable. If we view it as a 3-...
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Möbius strips with 3 twists to make a Klein bottle

I've been looking into Klein bottles and Möbius strips. What would happen if you took two "Möbius" strips with three twists in them, each oriented opposite eachother, and then connected the edges. ...
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Sphere with 1 disk removed and replaced with a Möbius strip

1) I read that a Klein bottle is in fact a sphere with 2 disks removed and replaced by Möbius strips. I find it hard to imagine how this constructs a Klein bottle. Any ideas how I can convince myself ...
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Proving homeomorphism in Möbius strip

Consider the Möbius strip defined by the following equivalence relation on the subspace $[0,1]\times]-a,a[$ of $\mathbb{R}^2$: $$(x,y)\sim (x',y')\implies (x,y)=(x',y')\vee|x'-x|=1\:\text{and}\:y'=-y$$...
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Orientation of a vector space

I want to open this question with a disclaimer, that I am not a native english speaker and I did not study linear algebra in english, and am simply trying to use words that I believe best translate ...
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Viewing Manifolds as Embedded in Euclidean Space

In learning differential topology I've been exposed with two methods of defining and working with manifolds: The more concrete but initially less general approach of Guillemin and Pollack, where all ...
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What is the area of a Möbius strip?

Yes, I know, it's not clear that we can define an area for a non-orientable surface etc. etc. So I'll try a more humble question: Following do Carmo, I parameterize the strip by $ \displaystyle x=(...
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Understanding Mobius bundle.

I'm trying to solve the following question: Consider the vector bundle $$E:= [0,1] \times \mathbb{R}/\sim \quad \to S^1$$ where $\sim$ is the equivalence rleation $(0,t) \sim (1,-t)$ for ...
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Matrices for symmetries of Möbius band

Taken from Groups and Symmetry by M.A. Armstrong, question 18.8: Setup: A 5x1 rectangular strip of paper is marked off on both sides into 5 unit squares. The two ends of the paper are then put ...
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Which is the function that the zero-set of which is Möbius strip?

Many surfaces can be expressed by implicit functions. As described later, the spherical surface, the side surface of the cylinder, and the torus can be expressed by implicit functions. However, I have ...
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Why is the Möbius strip not a product manifold despite being parameterizable as $(\theta, h)$?

I was reading this† lecture note and came across to the section on fibre bundles. On page 2 they mention that there is no unambiguous and continuous way to write a point $m$ on a Möbius strip $M_0$ ...
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Is a crosshandle homeomorphic to a Klein bottle?

I am aware that a Klein bottle is homeomorphic to two Möbius bands, and by Conway's zip proof a crosshandle is homeomorphic to two crosscaps. Now, since you can think of a crosscap as a Möbius band ...
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Why does the boundary of mobius strip wrap twice around core circle but not any other line?

A space $X$ deformation retracts onto a subspace $A$ if there exists continuous map $F:X\times [0,1]\rightarrow X$ such that $F(x,0)=x,F(x,1)\in A,F(a,t)=a$ $\forall a\in A$. The mobius strip ...
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How to prove that the Möbius band has geodesics?

In my class of Differential Geometry, the teacher defined geodesics as follows: A regular curve on a regular surface, denoted as $\gamma:I\subset\Bbb{R}\to S$, ($S$ is the surface) is a geodesic if, $...
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normal vector can not constructed at a point of a surface?

In page 66, chap 9 of the book "classical mechanics point particles and relativity" of Walter Greiner, say: "A surface for which a normal vector may be constructed at any point is called orientable." ...
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Why It's said to be that when a Flatlander makes a turn around a Möbius Strip, their internal organs are reversed, while they turn upside down?

I mostly hear that a flatlander becomes their mirror counterpart when they make a turn inside it. Though for that to happen, they need to be turned upside down. Does it not make a difference when they ...
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Mobius strip-an intuition

I do not follow how in the snippet below in the example 7.3 the space we are taking the quotient of is the whole of $[0,1]\times\mathbb{R}$? I mean, the diameter of the Mobius strip is bounded, but $\...
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Possibility of representing the Möbius strip by ways other than specifying a parameterization?

I have looked up some texts, they all represent the Möbius strip by ways of specifying a parameterization. This is the only way how the Möbius strip is represented founded in the Wikipedia currently. ...
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Knots on $\mathbb{R}P^2$ and on n-half-twist Möbius strips

Can you embed a non-trivial knot on the surface of $\mathbb{R}P^2$? I know we can on a torus, and we can't on a 2-sphere. My hand-wavy argument/intuition based on the little I know from self study of ...
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When is an open subset of an orientable manifold an orientable submanifold?

I am trying to complete problem $4$ from the introductory chapter of Riemannian Geometry by do Carmo. The question asks us to show that the projective plane $P^2(\mathbb{R})$ is non-orientable, first ...
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The image $M=\gamma(S)$ is a Mobius Strip, what are the missing points of the parameterization of the interior and what is another parameterization?

Let $$S=\{(r,\theta) | -\frac12 <r<\frac12 , 0\leq\theta<2\pi \}$$ and $$S'=\{(r,\theta) | -\frac12 <r<\frac12 , 0<\theta<2\pi \}$$ so $S'$ is the interior of $S$. Define $\gamma ...
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Do parallel, angle, triangle, area etc still apply in Mobius band?

Normal geometry concepts, such as parallel, angle, area, triangle, do they still apply in Mobius band? If not, in which case will they fail to do so? For example, what would three lines on a Mobius ...
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possible number of sheets for a Moebius band covering

Let M be the Moebius band, identified by the quotient of $[0,1]\times [0,1]$ by the equivalence $(x,0) \sim (1-x,1)$. Let $p: M\to M$ be a covering and $n$ its number of sheets. Find the possible ...
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is every path-connected covering of the Moebius strip a Galois cover?

Let $p : E → M$ be a covering of $M$ the Moebius Sttrip such that $E$ is path connected. Is this a Galois covering? My intuition is there must be some non locally path connected coverings that are ...
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Mobius strip with constant negative curvature

Is there any simple model of the Mobius strip with a constant negative Gaussian curvature? There is an example on Wikipedia (https://en.wikipedia.org/wiki/M%C3%B6bius_strip#Open_M%C3%B6bius_band), but ...
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Twisted Ring homeomorphic to Möbius Band?

Is the following parameterized surface homeomorphic to a Möbius Band? ...
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Twisting the unit square n times before gluing( 2.1.6 in G&P).

The question is given below: I have made a Mobius band with a paper and twisted it 3-times but I could not describe what I see it may be a 3 knot shape, could anyone give me a hint for solving that ...
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Group action of $\mathbb Z$ on infinite strip is homeomorphic to the Mobius Band

I am trying to prove that: Given $X=R×[-1, 1]$ and the action of $\mathbb Z$ as $m(x,y)=(x+m, (-1)^m y)$, prove that the space $X/Z$ is homeomorphic to the Moebius Band. Since there is no ...
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1answer
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Exercise 10. Groups and Covering spaces. Lima

Let $X$ be the space obtained from the sphere $S^2$ by gluing the north pole to the south pole, let $Y=\mathbb{R}^3-S^1$, where $S^1=\left\{(x,y,0)\in\mathbb{R}^3:x^2+y^2=1\right\}$ and let $Z$ be ...
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Which surface is homeomorphism to mobius strip?

I'm a bit confused since I read many versions of this. First of all, I couldn't understand how to define homeomorphism. Intuitively, I think that if I can deform an object, without tearing it, to ...
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1answer
333 views

Paradromic rings and Mobius strip

I'm working on a project about the differences between the original Möbius strip, a strip with an additional even number of half-twists, and a strip with an additional odd number of half-twists. This ...
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365 views

What is the 'center circle' of a Mobius Band?

What is the 'center circle' of a Mobius band? The question I am working on asks me to cut (literally) a Mobius band in half 'along its center circle.' What exactly does this mean? I know the plane ...