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I read a lot of historical articles that doubt the possibility that Gauss saw in the pseudosphere the realization of hyperbolic geometry; that geodetic triangles on the pseudosphere obey the same trigonometric rules valid in hyperbolic geometry (that can be modelled by the hyperbolic plane). Even proponents of the view that Gauss was in fact at full possesion of non-euclidean geometry, like Felix Klein (see: https://projecteuclid.org/download/pdf_1/euclid.bams/1183417390), doubt whether or not Gauss saw this connection, despite that to my opinion it's one of the more obvious conclusions of hyperbolic geometry.

I suspect he did see this connection, simply because he possesed all the essential elements needed in order to make this conclusion: he proved Gauss-Bonnet theorem for geodetic triangles on any surface, from his differential geometry he developed the equations of the pseudosphere (he called it "counterpart of the sphere") - the first example of a surface with constant negative curvature, and in his 1832 letter to Janos Bolyai he provided a synthetic proof of the angular deficit theorem in the hypothetic non-euclidean geometry (hyperbolic geometry). So if he knew both the angular deficit theorem of hyperbolic geometry and the Gauss-Bonnet theorem of differential geometry, and also had in hand an example of a surface with constant negative curvature, why there is still a long way to make this cnclusion?

I ask this question simply because i want to gain additional understanding of the subject of hyperbolic geometry. It's a simple fact that people tend to believe what they want to believe, and i personally have huge admiration for Gauss (so i preffer to believe he did make this conclusion), but i still want to gain better insight into this riddle.

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    $\begingroup$ Not that I could answer, but: why not cite at least some of this “lot of historical articles”? $\endgroup$ – Francois Ziegler Jan 15 '18 at 17:29
  • $\begingroup$ O.k i'll cite them; i don't remember all of them, but they were at least 5 or 6 in number. $\endgroup$ – user2554 Jan 15 '18 at 19:14
  • $\begingroup$ Your assumptions are wrong: the pseudosphere is not a "conclusion of hyperbolic geometry", it is just one of the possible local models, and in no way a necessary part of the hyperbolic geometry. $\endgroup$ – Alexandre Eremenko Jan 16 '18 at 21:15
  • $\begingroup$ Alexandre Eremenko - i'll ask another question: what common properties do hyperbolic triangles and geodetic triangles on the pseudosphere have? $\endgroup$ – user2554 Jan 17 '18 at 9:19
  • $\begingroup$ Beltrami introduced the pseudosphere as a model of hyperbolic geometry in 1868, Gauss passed away in 1855, you may be confusing him with Hilbert who proved that the pseudosphere is not extendable to the entire hyperbolic plane, or with Lambert who vaguely mentioned a hypothetical "imaginary sphere". The argument connecting hyperbolic and spherical trigonometries was made explicitly by Lobachevsky, it can be made into a proof of existence of hyperbolic geometry without constructing a model, see Rodin's Did Lobachevsky Have A Model. $\endgroup$ – Conifold Jan 18 '18 at 23:00
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Pseudosphere is not a realisation of hyperbolic geometry for two reasons: (1) it is not simply-connected; (2) it is not complete. Thus one cannot claim that there is a unique geodesic through any given pair of points, nor that any line can be indefinitely extended. In fact the hyperbolic plane cannot be embedded in Euclidean 3-space.

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  • $\begingroup$ Mikhail Katz - I don't fully understand your comment, but according to what i read the pseudosphere is a portion of the hyperbolic plane, not a complete model of of it, and that because it has singularity. But at least in parts of it the laws of hyperbolic trigonometry hold. That was the intention behind my question. $\endgroup$ – user2554 Jan 16 '18 at 17:43
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    $\begingroup$ @user2554, this is an error. The pseudosphere is not a portion of the hyperbolic plane. Only its universal cover (a different surface) can be identified with a portion of the hyperbolic plane. Alternatively, one can say that a portion of the pseudosphere can be identified with a portion of the hyperbolic plane. $\endgroup$ – Mikhail Katz Jan 16 '18 at 17:46
  • $\begingroup$ O.k probably i have a mistake (a fundamental misunderstanding). So i'll ask another question: what common properties do hyperbolic triangles and geodetic triangles in the pseudosphere have? $\endgroup$ – user2554 Jan 16 '18 at 17:51
  • $\begingroup$ If the perimeter of the geodesic triangle on the pseudosphere is a contractible curve on the pseudosphere then the triangle is isometric to a triangle in the hyperbolic plane, but otherwise it is not. $\endgroup$ – Mikhail Katz Jan 17 '18 at 9:41
  • $\begingroup$ @user2554: Katz uses "complete" in the technical, precise sense of differential geometry - meaning that every geodesic is infinitely extendable. You seem to use "not a complete model" as a not very precise synonym for "local model". I suspect part of the misunderstanding lies there. That some space is a model of hyperbolic space is a global statement. In fact, it's a theorem of Hilbert that there is no twice differentiable immersed complete (!) surface in $\mathbb{R}^{3}$ having constant Gaussian curvature. That is, there is no model of hyperbolic space as a surface in $\mathbb{R}^{3}$. $\endgroup$ – Dan Fox Jan 18 '18 at 15:19

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