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I've read many times that hyperbolic geometry is geometry on a negatively curved surface, but when I try to research it online, I usually get things like the Poincaré disk or Beltrami-Klein disk, which don't feel intuitive (especially on how distances are defined in both those models).

The current simplest model I can find is the Minkowski hyperboloid model (and both the Poincare disk and Beltrami-Klein disk models can be derived as azimuthal projections of the hyperboloid model), but there are two main difficulties I have with it:

  • it relies on a "Minkowski metric" for deriving distance that is different from the Euclidean metric (and the paper doesn't seem to describe how one might discover this metric in the first place, except in the context of general relativity)
  • it uses a hyperboloid of two sheets, which is a positively curved surface in 3D Euclidean space, not negatively curved

I also know of the pseudosphere model, which uses a negatively-curved surface but has complicated geodesics/lines as well as a video series called "Universal Hyperbolic Geometry" that looks promising in its simplicity but uses a completely different approach to hyperbolic geometry (and takes a very long to watch).

Because of this, is there a model of hyperbolic geometry that resolves all of these "problems" (i.e. it uses Euclidean metric, a negatively-curved surface, simple geodesics/lines, and a familiar geometric framework), or is it impossible?

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  • $\begingroup$ Are you familiar with the difference between intrinsic and extrinsic curvature, or for surfaces, the difference between the first and the second fundamental form? I am asking because the requirements from the last paragraph (Euclidean metric, negatively-curved surface, ...) seem to indicate some confusion about this... $\endgroup$
    – AlexD
    Apr 10 at 17:13
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    $\begingroup$ In the last paragraph, I mean an extrinsically negatively-curved surface being used to described an intrinsically negatively-curved geometry. $\endgroup$ Apr 10 at 17:39
  • $\begingroup$ Alright. Then see Somos' answer below. On a side note: you might also want to take a look at the half-plane model. I think it is quite intuitive and simple. $\endgroup$
    – AlexD
    Apr 10 at 17:56
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You asked

Because of this, is there a model of hyperbolic geometry that resolves all of these "problems" (i.e. it uses Euclidean metric, a negatively-curved surface, simple geodesics/lines, and a familiar geometric framework), or is it impossible?

and the simple answer is it is not possible. The reason is that what you want can only be done for a geometry that has greater curvature than the space it is embedded in. Thus, in Euclidean space of zero curvature we can have nice models of positively curved surfaces. If we lived in a negatively curved space, then we could have nice models of surfaces of greater curvature such as the Euclidean plane. However, we still could not have nice models of surfaces with more negative curvatures.

One reason, among others, for this situation is that the perimeter of a circle expands linearly as the radius increases in Euclidean spaces. In hyperbolic spaces, the perimeter expands essentially exponentially. Thus, there is no room in Euclidean space to contain all of the perimeter in a nice way without compromises.

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  • $\begingroup$ Seems to make sense; for a follow-up question, why does the Minkowski hyperboloid model use a two-sheeted hyperboloid instead of a one-sheeted hyperboloid? $\endgroup$ Apr 10 at 21:53
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    $\begingroup$ @coolcomputery Because the one-sheet has the wrong connectivity, similar to a cylinder. Topologically, disk and cylinder are not homeomorphic. $\endgroup$
    – Somos
    Apr 10 at 21:57
  • $\begingroup$ But is it still possible for a one-sheet model to satisfy the axioms of hyperbolic geometry and support a similar distance metric, even though such a model couldn't be projected to a disk? $\endgroup$ Apr 10 at 22:10

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