# 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...

30 questions
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### 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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### 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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### 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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### 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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### 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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### A non-orientable surface $S$ such that $T_pS$ is time-type.

I'm currently taking a course in Semi-Euclidean (Semi-Riemannian) Geometry and am given the task to give an example of a non-orientable surface $S$ in $\mathbb{R}^3_1$ which is time-type. With the ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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?
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### 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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### 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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### 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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### 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 ...
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?