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I have been doing some outreach work on conveying notions of hyperbolic space to people with limited math backgrounds. One idea I like to use is to talk about the hyperbolic plane as being like the opposite of the sphere. For instance, if I start with a circle (which is flat), then I shrink the boundary down to a point, it will bulge out into a sphere. So do the opposite: start with a circle and expand the boundary out to something infinite. This can be used to give some intuition about the classical models for the hyperbolic plane (upper half-plane, Poincaré disk, Klein disk, hyperboloid). In each of those models, one can think of it as having a boundary that is a circle of infinite radius, in different ways.

An audience member asked me an interesting question: why don't we use the hyperbolic paraboloid (familiar from calc 3) as a model for the hyperbolic plane? It captures this notion of a boundary needing to bulge out in different directions as one moves away from the center, like a Pringles potato chip or like choral. I'm wondering if anyone has ever made that idea precise, or if there is some reason why that can't be done. Is there a formula for a metric on the hyperbolic paraboloid that makes it a nice model for the hyperbolic plane? Is there some useful geometric interpretation for this? Are there interesting applications?

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j0equ1nn
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    Well, the hyperbolic paraboloid has non-constant curvature, when you consider the metric induced from $\Bbb R^3$. So you would have to find a motivation for a new metric there. And since that paraboloid is diffeomorphic to $\Bbb R^2$, you might as well work in the plane (which does not seem too interesting, since we already have Poincaré's half-plane). – Ivo Terek Jul 20 '18 at 19:05
  • I think that's what stopped me from having a good answer. On the other hand, I like the idea that the "center" of the hyperbolic paraboloid might, in some sense, capture what it "looks like" to stand a point in the hyperbolic plane and look around. But I'm not sure there's anything there that can be made rigorous constructively. – j0equ1nn Jul 20 '18 at 19:07
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    Well, if $X(u,v)=(u,v,u^2-v^2)$, then $K(X(0,0))= -4$, so the curvature of the hyperbolic paraboloid remains negative in a neighborhood of the "center" by continuity. It is an hyperbolic point in the sense that the surface locally intersect the half-spaces determined by tangent plane at the "center". I don't know how further can we go. The curvature of the paraboloid being non-constant is a great problem. – Ivo Terek Jul 20 '18 at 19:17

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I think the hyperbolic paraboloid is a great model to build intuition about what negative Gaussian curvature is, and how it would interact with things like angle sum. You might also look at images for “crocheted hyperbolic plane”, or create one yourself, to build some even stronger intuition how adding more matter than what would be needed for a planar setup will automatically lead to curvature, and how that curvature might have different embeddings which do not affect the intrinsic metrics.

As comments pointed out, the curvature is not constant, so at that point it's a poor model for the hyperbolic plane. Other models like the tractricoid or Amsler's surface have constant negative curvature, so they can represent a portion of the hyperbolic plane with accurate metrics.

But the key problem with all these isometric 3d models is that they at best only represent a portion of the plane, not all of it. They are ill suited to studies of the global structure of the hyperbolic plane. So at some point you have to give up hoping for an isometric embedding, and get used to dealing with somewhat non-intuitive metrics.

As for the first paragraph, about building intuition by speaking about a circle infated to infinite radius, I'm not sure I'd be happy with that. I can see many ways how this could be misread. But that's my personal impression; if your students feel that it helped them, that's good enough in my opinion.

MvG
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  • Regarding the inflated boundary part, I think I know what you mean. It could be interpreted as just making a flat plane. The info depends a lot on weird drawings and gesturing, which aren't a great substitute for rigor, but I do feel people went away with some idea of what "hyperbolic" means. – j0equ1nn Jul 20 '18 at 20:12