There is a question on this site about the distinctions between the full-twisted Mobius band and the cylinder, but I would like to ask something different, so I start a new question. Let us call $C$ the standard cylinder embedded in $\mathbb{R}^3$, and call $F$ the full-twisted Mobius band (aka. the Mobius band with two […]

I understand that if you were to walk on the surface of a Möbius strip you would have the same perspective as if you walked on the outer surface of a cylinder. However, would it be possible for someone to determine whether they were on a Möbius strip or cylinder.

We can distinguish between a (closed) Mobius strip and ‘regular’ (untwisted) strip by examining the set of points which have no neighborhood homeomorphic to a disk (intuitively, the ‘boundary’ of the strip). For a Mobius strip, this set is connected, while for an untwisted strip the set has two connected components. However, the above criteria […]

Consider the Klein bottle (this can be done by making a quotient space). I want to give a proof of the following statement: The Klein Bottle is homeomorphic to the union of two copies of a Möbius strip joined by a homeomorphism along their boundaries. I know what such a Möbius band looks like and […]

I would like to have Mathematica plot a “thickened Möbius strip”, i.e. a torus with square cross section that is given a one-half twist. Ideally, I would like this thickened Möbius strip to be transparent with a (non-thickened) solid Möbius strip sitting at its center; here is the best approximation I could draw by hand […]

Does the concept of a Möbius torus make sense: taking a cylinder (instead of a rectangle as in the case of the Möbius strip) and twisting it before joining its ends? Or will the resulting twisted torus be indistinguishable from the normal torus in any relevant respect? [This equivalent to the well-known Möbius strip should […]

Why does the result of cutting a Möbius strip down the middle lengthwise have two full twists in it? I can account for one full twist–the identification of the top left corner with the bottom right is a half twist; similarly, the top right corner and bottom left identification contributes another half twist. But where […]

I am trying to understand the notion of an orientable manifold. Let M be a smooth n-manifold. We say that M is orientable if and only if there exists an atlas $A = \{(U_{\alpha}, \phi_{\alpha})\}$ such that $\textrm{det}(J(\phi_{\alpha} \circ \phi_{\beta}^{-1}))> 0$ (where defined). My question is: Using this definition of orientation, how can one prove […]

Intereting Posts

Ring with finitely many zerodivisors
implicit equation for elliptical torus
$5^n+n$ is never prime?
Showing $\sum_0^n {n\choose k}(-1)^k \frac{1}{2k + 2} = 1/(2n + 2)$
An application of Jensen's Inequality
Prove there is no strictly increasing function $f$ from irrationals to reals.
Intuition behind Conditional Expectation
Given a prime $p\in\mathbb{N}$, is $A=\frac{x^{p^{2}}-1}{x^{p}-1}$ irreducible in $\mathbb{Q}$?
A Riesz-type norm-preserving and bijective mapping between a Banach space and its dual
Finding the derivative of $x^x$
Extreme points of unit ball of Banach spaces $\ell_1$, $c_0$, $\ell_\infty$
Bijection between $$ and $
Formula for the sum of $\ n\cdot 1 + (n-1)\cdot 2 + … + 2 \cdot (n-1) + 1\cdot n$
Problem similar to folland chapter 2 problem 51.
Units in a ring of fractions