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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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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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1answer
183 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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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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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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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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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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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 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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51 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 ...