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I was shown this paper which proves that hyperbolic $H^n$ space can be isometrically embedded in Euclidean $E^{6n-6}$ space.

While I am able to understand pieces of the proof, I am not an advanced mathematician and am having trouble pulling out much usable information from the paper.

For the simplest case of the hyperbolic plane $H^2$ being isometrically embedded in $E^6$, is it practical to do so? The motivation behind this question is to acquire a set of usable equations in computing.

Most importantly, I need a way to test that a point in $E^6$ lies on $H^2$. This is analogous to the equation $x^2 + y^2 + z^2 = 1$ that can be used to verify a point in $E^3$ lies on the surface of the unit sphere, and thus in $S^2$.

I recognize that "practical" is a subjective word to use. But broadly speaking, if such equations involve solving sixth degree polynomials or the use of the smooth-step function in the paper's appendix, then I would say they are "impractical."

I am aware that is it common to compute points in $H^2$ using the hyperboloid model. However it would be useful for some algorithms (in particular noise generation) if I could isometrically embed the entire plane into a Euclidean space.

Tim Morris
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    "The motivation behind this question is to acquire a set of usable equations in computing." - what are you planning to compute that is not already computable using standard models of the hyperbolic plane? – Moishe Kohan May 25 '23 at 03:43
  • @MoisheKohan As I specified in the question, it would be useful for noise generation functions (such as simplex noise). But even if something is "already computable" using standard models, that does not satisfy the simple curiosity I have. It would be interesting to explore algorithms from this hypothetical model. – Tim Morris May 25 '23 at 03:46
  • Ok, good luck with that, however, I think you will be better off not following this route. – Moishe Kohan May 25 '23 at 03:53
  • @MoisheKohan I appreciate the warning, I suppose? A lot of mathematics is explored for simple curiosity. It is not as if every endeavor must have provable value before being explored. – Tim Morris May 25 '23 at 03:56
  • I do not understand your “where all axes are perpendicular to one another.” This is true in Minkowski space, as well. – Ted Shifrin May 25 '23 at 03:57
  • @TedShifrin I suppose that was poorly phrased. I'll simply remove that. It was not necessary for the core of the question anyway. – Tim Morris May 25 '23 at 03:59
  • So, for your purposes, what is wrong with the pseudosphere in $\Bbb R^3$? – Ted Shifrin May 25 '23 at 04:00
  • The pseudosphere doesn't cover the entire hyperbolic plane. Plus, if you will forgive my non-rigorous language, the sampling of points gets "stretched out" as you get further away from the origin in both the pseudosphere and in Minkowski space. If the same is true for embedding $H^2$ in $E^6$, I am unaware. – Tim Morris May 25 '23 at 04:06
  • You're right that much mathematical exploration is driven by simple curiosity, and I'm confident that this is at least part of what drove the search for an isometric embedding of $\mathbb H^n$ into some Euclidean space. But you seem to be asking for more, namely a simpler proof of the existence of this embedding. That is asking for a lot. – Lee Mosher May 25 '23 at 14:31
  • @LeeMosher I (perhaps wrongly) assumed that since it was proven that you can do it, that there exist equations for it. I didn't consider that simple equations imply there exists a simpler proof. I suppose in that case the answer to my question is "no," then. However, would it be appropriate to leave this question open? I can edit it to clarify that I'd like to see even "impractical" equations if they ever surface. – Tim Morris May 25 '23 at 16:26
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    Functions (such as an isometric embedding $\mathbb H^2 \mapsto \mathbb E^6$) need not be specified by equations expressed in elementary terms. Much of mathematical analysis (i.e. "advanced advanced calculus") is devoted to powerful abstract methods for finding useful functions even without the ability to express them in elementary terms. – Lee Mosher May 25 '23 at 17:37
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    You might be familiar with at least one such abstract theorem, if you ever took an undergraduate course in ordinary differential equations: the existence and uniqueness theorem for solutions to linear ODEs with initial conditions. I would even say that the existence theorem for indefinite integrals, in an ordinary calculus class, is also a nice source of lots of functions that do not have elementary expressions. – Lee Mosher May 25 '23 at 17:38
  • @LeeMosher I took an intro to diff-eq class, yeah. The comparison makes sense and I am frustrated (at math, not you) that this is the case, haha. Thanks for the info! Would it be appropriate for you to leave a short answer outlining the explicit vs implicit nature of the problem, so I can mark it as correct? I'd rather not leave the question open nor simply delete it. – Tim Morris May 25 '23 at 17:50
  • @TimMorris "the sampling of points gets "stretched out" [...] in Minkowski space" - I don't think this is the case (though I'm not sure exactly what you mean). You might be able to use some of the ideas in this answer of mine. – mr_e_man Jun 12 '23 at 18:47

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Let me summarize my comments.

It is a mathematical fact of life that functions need not be expressed in explicit, simple, elementary terms. One encounters this in undergraduate mathematics theorems such as: the existence of indefinite integrals; the implicit function theorem; the existence/uniqueness theorem for solutions of linear ODE's with initial conditions.

As mathematical analysis has matured over the decades (centuries?), powerful, clever tools for constructing functions have emerged. Of course simpler constructions are better, but they might not be easy to find at all, and you might have to settle for a complicated construction.

I suspect this is the case with isometric immersions of $\mathbb H^n$ into Euclidean space, but of course there's always the choice of working hard and reading and understanding the paper you linked.

Lee Mosher
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  • "There is aways the choice of working hard and reading and understanding the paper" To be clear, I never expected someone do the work for me. I doubt that is what you were insinuating -- but I just wanted to clarify my intent was to see if the work had already been done, so to speak. – Tim Morris May 25 '23 at 18:47
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    What I meant was not either of those. It was that with a lot of work reading the paper, one might obtain an understanding of a probably verrrrry complicated method of writing down the embedding, *which* might nonetheless be useful... who knows? – Lee Mosher May 25 '23 at 21:03