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?


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: $f : (-\frac{\pi}{2}, \frac{\pi}{2})^4 \to \mathbb{R}^4$ $$f(x_1, x_2, x_3, x_4) = (\tan(x_1),\tan(x_2),\tan(x_3),\tan(x_4))$$

  • $\begingroup$ 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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