I've seen a few posts here that make certain claims that are related to Hilbert's theorem. For instance:

"I know that there is no complete surface embedded in $\Bbb R^3$ of constant curvature $-k$ for any $k$."

"However, as Hilbert showed us, the reverse is not true; we cannot embed the hyperbolic plane into Euclidean 3-space."

"$H^2$ does not isometrically embed in $R^3$ (Hilbert's theorem)."

However, these should only apply to smooth ($C^\infty$) embeddings. Nash-Kuiper, particularly, guarantees the existence of an isometric $C^1$ embedding, since the Klein disk model is a short map from $H^2 \to \Bbb R^3$.

This was noted in the answer to this MO question.

There's also this answer to the same question, which says there's no $C^2$ embedding.

But then on the other hand, this page on hyperbolic crochet at Cornell claims that the crochet models are not $C^1$ and can be extended indefinitely, which would appear to violate Hilbert's theorem.

At this point, I'm very confused about what is and isn't allowed regarding isometric embeddings of $H^2$ into $\Bbb R^3$. Obviously no infinitely differentiable isometric embedding exists, but there are obviously $C^1$ embeddings of the whole space. Furthermore, there are also isometric embeddings of compact subsets of $H^2$ into $R^3$, such as the crochet models, and it's not clear what differentiability restrictions there are for that.

Is there some way to better understand the slew of somewhat-contradictory statements made above? What, exactly, is and isn't allowed regarding isometric embeddings of $H^2$ into $R^3$?

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    $\begingroup$ I have not looked back at the proof but all you should need is a complete $C^2$ submanifold with constant negative curvature - no boundary. You need $C^2$ so that the submanifold's curvature is, well, defined, and also the same as that of $H^2$ itself. $\endgroup$
    – user98602
    Nov 19, 2015 at 1:39
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    $\begingroup$ Quoting the linked page, We will show that the surfaces described here can be extended indefinitely and that the intrinsic geometry of these surfaces is hyperbolic geometry; and, they appear to not be $C^{1}$ embedded. (I read this to mean "not even $C^{1}$".) That is, everything you've linked appears to be consistent with the assertions: 1. There exists a $C^{1}$ isometric embedding of $H^{2}$ in $E^{3}$. 2. There does not exist a $C^{2}$ isometric embedding of $H^{2}$ in $E^{3}$. $\endgroup$ Nov 19, 2015 at 2:11
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    $\begingroup$ Incidentally, there certainly exist real-analytic isometric embeddings of (sufficiently small) hyperbolic disks in $E^{3}$. Offhand I don't know the maximum hyperbolic radius, however. $\endgroup$ Nov 19, 2015 at 2:15
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    $\begingroup$ A crocheted model is a discrete model (with stitches as "atoms", and hyperbolicity achieved by "packing in" more stitches than Euclidean crocheting accommodates), analogous to a polyhedral model (which is also not differentiable at the vertices, where the curvature is concentrated). Perhaps that's what the page author meant? $\endgroup$ Nov 19, 2015 at 11:52
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    $\begingroup$ Even ignoring constraints of neighborliness (come to think of it), the number of stitches that can be physically packed into a given Euclidean $3$-ball grows polynomially with the radius, while the number of stitches in a given crocheted patch grows exponentially with the hyperbolic radius, and therefore with the Euclidean radius of the model. At some point your crocheted model grows into a solid mass of stitches, prohibiting further crocheting. :) $\endgroup$ Nov 22, 2015 at 1:09

1 Answer 1

  1. A close look at the hyperbolic crochet article that you linked reveals very little mathematical content to it. The link does not even propose a Lipschitz isometric embedding of the hyperbolic plane into $E^3$. This might be a good mathematical PR article but not more than that. No wonder, one gets confused trying to read it. Can this paper be made precise and yield an actual mathematical theorem? Who knows (the paper is by now 16 year old), until then I suggest it should not be used as a reference.

  2. The first proof of nonexistence of a $C^2$-smooth isometric immersion of a hyperbolic plane into $E^3$ is due to Efimov:

    N. V. Efimov, Impossibility of a complete regular surface in Euclidean 3-Space whose Gaussian curvature has a negative upper bound, Doklady Akad. Nauk. SSSR 150 (1963), 1206–1209 (Russian); Engl. transl. in Sov. Math. (Doklady) 4 (1963), 843–846.

    N. V. Efimov, Differential criteria for homeomorphism of certain mappings with application to the theory of surfaces, Mat. Sb., Nov. Ser. 76 (1968) (Russian); Engl. transl. in Math. USSR (Sbornik) 5 (1968), 475–488.

In fact, he proves an even stronger result:

Theorem. Suppose that $M$ is a simply connected complete Riemannian surface of sectional curvature $\le k<0$. Then there is no $C^2$-smooth isometric immersion of $M$ into $E^3$.

This is a vast generalization of Hilbert's theorem on nonexistence of $C^\infty$-smooth isometric immersions of the hyperbolic plane into $E^3$.

  1. The pseudosphere provides a $C^\infty$-smooth isometric immersion of a horodisk in the hyperbolic plane into $E^3$. In particular, hyperbolic disks of arbitrarily large radius can be isometrically immersed into $E^3$.

  2. It appears to be an open problem: What is the largest radius of a hyperbolic disk that can be $C^2$-smoothly isometrically embedded into $E^3$.

  3. As a special case of Nash-Kuiper isometric embedding theorem, there exists a $C^1$-smooth isometric embedding of the hyperbolic plane into $E^3$.

  4. For further reading:

Qing Han and Jia-Xing Hong, Isometric Embedding of Riemannian Manifolds in Euclidean Spaces, AMS Mathematical Surveys and Monographs, vol. 130, 2006.

  • $\begingroup$ What Is a PR article? $\endgroup$ Dec 3, 2020 at 15:01
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    $\begingroup$ @AntonioJPan: PR= Public Relations $\endgroup$ Dec 3, 2020 at 17:53

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