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I'm looking for an explicit diffeomorphism between the four-dimensional euclidean space $\Bbb{R}^{4}$ and the four-dimensional open cube. I wonder whether there is a simple looking map, with simple looking derivatives (I need to induce a metric on the cube from the space, and I'd prefer it didn't look terribly complicated at the end). Is anybody able to help me with this?

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We can use the result proved in this question to build one. There it is shown that $f : (-\frac{\pi}{2}, \frac{\pi}{2}) \to \mathbb{R}$ defined by $$f(x) = \tan(x)$$ is a diffeomorphism. To obtain the function you are looking for, use four copies of that one: $f : (-\frac{\pi}{2}, \frac{\pi}{2})^4 \to \mathbb{R}^4$ $$f(x_1, x_2, x_3, x_4) = (\tan(x_1),\tan(x_2),\tan(x_3),\tan(x_4))$$

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  • $\begingroup$ Thank you so much, it looks perfect to me. I'll build the solution and get back to you. $\endgroup$ – Giorgio Comitini Jul 27 '15 at 12:41
  • $\begingroup$ Yes, this is definitely what I was looking for. Simple to write and even easier to differentiate. Thank you again! $\endgroup$ – Giorgio Comitini Jul 27 '15 at 12:50
  • $\begingroup$ @GiorgioComitini Happy to help! $\endgroup$ – muaddib Jul 27 '15 at 12:51

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