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Consider the following question:

"Can the hyperbolic plane $(\mathbb{R}^2, g_H)$ be isometrically immersed in three dimensional Eulidean space$(\mathbb{R}^3, g_{flat})$?"

I believe the answer to this question is no. Can someone give me a reference for this theorem (in particular I want to look at the details of the proof and understand why this is not possible).

My second question is as follows: I assume the answer is no if one is asking for smooth (i.e. $\mathcal{C}^{\infty}$) immersion. Is anything known if we relax this condition? More precisely, asking for an immersion is asking whether a certain pde has a solution. What happens if I am just looking for some "weak" solution to this pde?

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    $\begingroup$ This is known as Hilbert's theorem en.wikipedia.org/wiki/… . If I recall correctly there is a discussion of this as well as other "no-go" theorems in the book "Isometric Embedding of Riemannian Manifolds in Euclidean Spaces" by Qing Han and Jia Xing Hong which you have cited in another question. See also this question mathoverflow.net/questions/111101/… $\endgroup$
    – j.c.
    Aug 20, 2013 at 16:39
  • $\begingroup$ Thank you for the wikepedia reference. I do have the book you mentioned; I am not certain however if he discusses any notion of a "weak" solution for an immersion there. $\endgroup$
    – Ritwik
    Aug 20, 2013 at 16:43

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The Nash-Kuiper embedding theorem applies here, as the obvious smooth topological embedding of the hyperbolic plane as the unit disk in 3-dimensional Euclidean space decreases distances. Therefore there is a $C^1$ isometric embedding. I don't know of a stronger result.

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Without the deep theorem of Nash-Kuiper, it is easy to construct and visualize a bi-Lipschitz embedding: think of those Dutch collars of the XVII century like this: https://en.wikipedia.org/wiki/File:Rembrandt,_Portret_van_Haesje_v.Cleyburg_1634_2.jpg

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Klotz Milnor proved in 1972 that there is no $C^2$ isometric embedding of $H^2$ in $R^3$ so Nash–Kuiper cannot be improved: Zbl 0236.53055.

A practical realisation of an embedding is here: Crocheting the hyperbolic plane by David W. Henderson and Daina Taimina.

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    $\begingroup$ Note that the theorem is really due to Efimov (who proved that there is complete, $C^2$ surface in $R^3$ with curvature $K\leq -\epsilon<0$ -- this applies in particular to the hyperbolic plane). The paper by Klotz-Milnor is a (very nice) exposition of Efimov's result. $\endgroup$
    – Jean-Marc Schlenker
    Aug 27, 2013 at 19:22
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    $\begingroup$ you mean of course there is NO such surface $\endgroup$
    – ThiKu
    Aug 28, 2013 at 2:45
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    $\begingroup$ Yes that's right of course! I don't see how to edit this comment now, but it should read "...that there is NO complete...". Thanks for the remark. $\endgroup$
    – Jean-Marc Schlenker
    Aug 28, 2013 at 11:36

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