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I know well that there is an isometric immersion of the hyperbolic plane in $\mathbb R^5$ due to Rozendorn and that the hyperbolic plane cannot be immersed isometrically in $\mathbb R^3$ due to Hilbert, also I think that the problem in $\mathbb R^4$ is still open. So questions arise that I can't answer:

  1. Are there surfaces with negative Gaussian curvature immersed isometrically in $\mathbb R^4$?
  2. Are there surfaces with negative Gaussian curvature with $K\leq const<0$ immersed isometrically in $\mathbb R^4$?
  3. Are there surfaces with constant negative Gaussian curvature immersed isometrically in $\mathbb R^4$?

Does anyone know where I can read more about the existence of complete surfaces with constant Gaussian curvature -1 immersed isometrically in $\mathbb R^4$?

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    $\begingroup$ Just a small correction. In view of the pseudosphere, please add “complete” to your Hilbert discussion. $\endgroup$
    – Ted Shifrin
    Aug 11, 2021 at 1:31
  • $\begingroup$ Yes, I'm taking the complete hyperbolic plane as otherwise I would have put a part of the hyperbolic plane which is immersed isometrically in $\mathbb R^3$ represented as the pseudosphere, for example. $\endgroup$
    – Zaragosa
    Aug 11, 2021 at 13:53
  • $\begingroup$ If you're OK with lower regularity maps, you can upgrade them to isometric embeddings, by Nash-Kuiper. You start by embedding your surface any old way by the hard Whitney theorem, and then use this result to find a nearby isometric embedding which is $C^1$. I wouldn't be surprised if some of these questions were open otherwise though. $\endgroup$
    – A. Thomas Yerger
    Aug 12, 2021 at 22:48

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