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How can I see inside the projective plane the Möbius band?

I need to know how the Möbius Band appears inside the projective plane.

I know it is easy using identifications and algebraic topology.

I want to use parametrization of both the inside and the outside manifolds now.

Narasimham
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stroncoso1
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1 Answers1

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The Projective Line $P^1R$ is sitting inside the projective plane $$P^2R=\left\{\left[x:y:z\right]\right\}$$ as $$P^1R=\left\{z=0\right\}$$ It is well known that $P^1R$ is homeomorphic to a circle. Now look at (edited:) some $\epsilon$-neighborhood of $P^1R$ for some positive $\epsilon$. You can check that is a nontrivial interval bundle over $P^1R$, so ist is a Möbius band.

user39082
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  • The condition $|z|<\epsilon$ is not well defined for a point $[x:y:z]\in P^2\mathbb R$, since $[x:y:z] = [\lambda x:\lambda y:\lambda z]$ for every nonzero real number $\lambda$. – Jack Lee Feb 17 '15 at 05:32
  • That's right, what I meant was: take a sufficiently small neighborhood of $P^1R$ (e.g. inside a chart). – user39082 Feb 18 '15 at 09:55
  • In fact it suffices to take out (a small neighborhood of) $\left[0:0:1\right]$ from $P^2R$. – user39082 Feb 18 '15 at 09:56