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Given a set $M = \{(x,y,z)\in \mathbb R^3 | x^2+y^2-z^2=1\}$ and a function $\alpha \colon M \to S^1 \times \mathbb R$, $$\alpha(x, y, z) = \left(\frac{(x,y)}{\sqrt{1+z^2}},z\right)$$

Is there a smart method of showing that $\alpha$ is a diffeomorphism? I already showed that $M$ is an immersed submanifold of $\mathbb R^3$.

Is it enough to show that it is a bijection, the map is smooth, and the jacobian is nowhere 0?

guest
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  • $\alpha$ looks like sterographic projection but with a one-sheet hyperboloid ($H_1$) taking the place of a sphere. – Jean Marie Nov 03 '16 at 17:34
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    A comment about terminology: the word you want in English is "map" (for the French term "application"). The English word "application" means something entirely different. – Jack Lee Nov 03 '16 at 20:46

1 Answers1

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Writing $\alpha(x,y,z) = (u,v)$ with $u \in S^1 \subseteq \mathbb{R}^2$ and $v \in \mathbb{R}$, we have $\alpha^{-1}(u,v) = (\sqrt{1 + v^2}u,v)$. Now just verify that both $\alpha$ and $\alpha^{-1}$ are smooth.

levap
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  • which charts and maps should I use to verify their smoothness? – guest Nov 03 '16 at 23:38
  • @JonathanBaram: Note that both your manifolds, considered as submanifolds of $\mathbb{R}^3$, have rotational symmetry around the $z$-axis. Hence, it will be the easiest to use charts that describe both as surfaces of revolution. – levap Nov 04 '16 at 00:53
  • could you develop what it means to have charts that describe as surface of revolution – guest Nov 04 '16 at 12:01
  • I showed that $M$ is an immersed submanifold of $R³$, cant I use this ? – guest Nov 04 '16 at 14:22