Questions tagged [non-orientable-surfaces]

For all questions about Möbius bands, Klein bottles, projective planes or surfaces built from these (via surgeries, gluings...), intersection or boundary problems, embeddings...

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Clarifying some details about Orientability of Surfaces using Vector Fields

For an orientable surface embedded in $\mathbb{R}^3$, we can properly define a normal vector field on it, and we can't do so on a nonorientable surface. On the other hand, there is a result saying ...
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36 views

Immersion of non-orientable surface in $\mathbb R^3$ with conditions on the height function

EDIT: The answer is trivially positive: see this answer. Can a non-orientable closed surface of odd genus be immersed in $\mathbb R^3$ so that the associated height function be of Morse-Bott type and ...
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1answer
21 views

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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1answer
33 views

¿Does a timelike non-orientable surface exist?

Consider the Lorentz-Minkowski space of dimension $3$, $\mathbb{L}^3 = (\mathbb{R}^3,\langle \: \cdot \: \rangle)$ $$ \langle u,v\rangle=u_1v_1 + u_2v_2 - u_3v_3 $$ We say that a surface $S$ is ...
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2answers
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Classification of surfaces

The Classification Theorem for surfaces says that a compact connected surface $M$ is homeomorphic to $$S^2\# (\#_{g}T^2)\# (\#_{b} D^2)\# (\#_{c} \mathbb{R}P^2),$$ so $g$ is the genus of the surface, $...
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37 views

Is there an intuitive way to see why $\mathbb{P}^2$ is non-orientable?

In the book of D. Chillingworth titled Differential Topology with a view to applications, at page 141, the author argues that [...] $\mathbb{P}^2$ can be described equivalently as the set of all ...
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1answer
59 views

Triviality of complexified tangent bundle of a closed surface

Does anybody know how to prove the following statement: The complexified tangent bundle $TS\otimes\mathbb{C}$ of a closed surface $S$ is topologically trivial iff the Euler characteristic $\chi(S)$ ...
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1answer
145 views

Klein bottle with constant curvature is flat

A torus (equipped with a Riemannian or Lorentzian metric) which has constant curvature must be flat because of Gauss-Bonnet theorem. Is it true that a Klein bottle (equipped with a Riemannian or ...
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1answer
64 views

What goes wrong with Stokes theorem if a surface is not orientable?

For the Möbius strip parametrized by $\{\sigma(\theta,r)=((1+r\sin(\theta/2))\cos\theta,(1+r\sin(\theta/2))\sin\theta,r\cos(\theta/2))\ \mid \\ (\theta,r)\in A=(0,2\pi)\times(-1/2,1/2) \}$ we get ...
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Given a free involution $i$ on a finite graph $G$, is there a minimal embedding of $G$ such that $i$ is facial?

This question comes from my own Bachelor-thesis work. I am exploring the 1-2-inifity conjecture, or Negamis conjecture, and I am trying to see what happens if one tried to extend Negamis result that a ...
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1answer
76 views

Are non-orientable manifolds necessarily compact?

If not, what is an example of a non-compact, open manifold that is non-orientable? So if non-orientability $\Rightarrow$ compactness, is there a theorem and what is the proof?
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An intelligent way to see if this surface is orientable

I am trying to do this excercise: \begin{array} { c } { \text { Let } M \text { be the regular surface in } \mathbb { R } ^ { 3 } \text { parametrised by } } \\ { X : \mathbb { R } \times ( - 1,1 ) \...
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Using fundamental polygons to prove $PR^2\# T^2 = PR^2\# PR^2 \# PR^2$.

Background Let $P^2$ denote the real projective plane, and $T^2$ the torus. These generate a monoid where the operation is the connected sum. This monoid is abelian, so for notational brevity, we ...
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277 views

Does this manifold have a name?

EDIT: In an attempt to not let the bounty go to waste, I will consider responses that give reasonable guesses of what the involved surfaces are, WITHOUT requiring parametrizations. While using ...
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3answers
856 views

Does the Gauss-Bonnet theorem apply to non-orientable surfaces?

I found statements of the Gauss-Bonnet theorem here, here, here, here, here, here, here, and here. None of them require that the surface be orientable. However, Ted Shifrin claims in a comment to this ...
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67 views

How to arrive at an equation of a Roman surface from three points (a triangle)

I have three points (for example, A( 1, 1, 0 ) B( 2, 1, 0 ) C( 1.5, 0, 0 ), origin at O( 0, 0, 1000 ) ), with which I am creating what I believe is a Roman surface ($x^2y^2 + y^2z^2 + z^2x^2 - r^2xyz =...
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1answer
109 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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2answers
85 views

Immersion of non-orientable manifold in a small orientable one

I was trying to prove the following fact: given a non orientable manifold $M$ of dimension $m$, $M$ is always contained in an orientable manifold of dimension $m+1$. I have gotten nothing out of it, ...
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52 views

Existence af a Frame on the Klein Bottle

I was trying to disprove the fact that there exist a global tangent frame on the Klein bottle, i.e. two global vector fields everywhere indipendent. Since my background involves no charachteristic ...
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61 views

Criterion for being in a non-orientable 3 manifold?

I'm trying to wrap my head around the concept of orientability as an intrinsic property of a manifold. Assume I'm in some (3-dim) manifold for which I'd like to decide its orientability; what could I ...
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1answer
129 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 ...
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1answer
214 views

Is there a way to prove algebraically that a Möbius strip is non-orientable?

I am doing my HL Maths coursework on non-orientability of surfaces and am trying to prove whether a möbius strip is orientable or not (of course it isn't) Is there a way to prove algebraically that a ...
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1answer
98 views

Example of a non-orientable 3-manifold

I was reading a paper and it was affirmed in there that $\mathbb{R}\mathbb{P}^2\times\mathbb{S}^1$ was a non-orientable 3-manifold. Does anyone knows how to prove it? if not, is there another (simple)...
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1answer
128 views

Does Day and Night on a Klein bottle have a steady state?

Place a $m \times n$ ($m,n \ge 3$) square grid on a Klein bottle. On each square, we select an arbitrary non-mirror symmetric marker, and arrange them on the Klein bottle in some way. This arrangement ...
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1answer
75 views

When you divide the real projective plane into two subsets, does it always have exactly one non-orientable component?

Let's say you divide the real projective plane into two subsets, are exactly one these subsets non-orientable? In particular, we will require that each subset $S$ is "nice" in the sense their common ...
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2answers
307 views

Non-orientable cover of a non-orientable surface

I was quite puzzled by the request of classifying all the $4$-covers of the connected sum of $5$ copies of $\Bbb R P^2$. For oriented covering space, the answer is well known: it's enough to consider ...
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344 views

Fixed points in mapping from Möbius strip to disk [Explanation or reference needed]

One of the most elegant demonstrations in topology is the proof of the inscribed rectangle problem (a solved variant of the unsolved inscribed square problem) which states that for any plain, closed ...
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2answers
2k views

Why is Klein bottle non-orientable?

I am doing the homework of differential geometry and encounter this problem: The Klein bottle $K^2$ is defined to be the identification space $$[0, 1] \times [0, 1]/{\sim}, \text{ where the ...
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1answer
308 views

A space that deformation retracts into the cylinder and Möbius band doesn't embed in $\Bbb R^3$.

Consider the Möbius band, and take the middle circle in it (so that it deformation retracts onto it). Glue the upper boundary of a cylinder through it. This gives a space that deformation retracts ...
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2answers
552 views

Klein bottle contains Möbius band

I read the following: "The Klein bottle contains a copy of the Möbius band". I assume this means that there is a subspace of the Klein bottle that is homeomorphic to the Möbius band. How do we obtain ...
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1answer
796 views

Torus/Möbius Band homeomorphism

Is a fattened Möbius Spiral Band homeomorphic to a Torus? (Due to the same Euler Characteristic $\chi$ ?) Are both non-orientable? Following (3D printable plastic) Torus has a square section that ...
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1answer
886 views

Orientable Surface Covers Non-Orientable Surface

I need to describe how a 4-genus orientable surface double covers a genus 5-non-orientable surface. I know that in general every non-orientable compact surface of genus $n\geq 1$ has a two sheeted ...
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1answer
456 views

Cell decomposition of non-orientable surfaces

I saw that a cell-decomposition of a genus g non-orientable surface is $D^0\cup D^1\cup ...\cup D^g$. Can anyone explain why this is true?