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Why would one study hyperbolic geometry? I am only aware of the motivation where you give axioms for elementary euclidean geometry and then start to wonder wether the parallel axiom is necessary. You then see that if you negate the axiom you get the hyperbolic space instead of the euclidean space. But if this were the only motivation then one might very well learn the construction of the space and then stop. Instead it is taught in elementary and differential geometry and this can't be only because the theorems are exotic if compared to the euclidean case.

I am mostly looking for mathematical motivations here, so what are relations to other fields, what are the advanced topics and such. Why is hyperbolic geometry of interest for a mathematician?

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    $\begingroup$ One would study Hyperbolic Geometry because it is stunningly beautiful? But that's just an opinion :) $\endgroup$ – imranfat Aug 26 '15 at 15:13
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    $\begingroup$ As one Blue to another: Because it's there! Also, because even some of the "easy" stuff about geometric objects remains unknown, so opportunities for discovery abound. For instance, only recently (2005) did we get a reasonably-attractive formula for the volume of a tetrahedron; and I beileve a number of results in my "Hedronometric Formulas for a Hyperbolic Tetrahedron" are new. This makes exploration all the more satisfying ... and fun! $\endgroup$ – Blue Aug 26 '15 at 15:42
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First, we should note that a very similar question has already been asked here, and several interesting answers were given. But because the question keeps coming up, I'm going to go out on a limb and suggest that there still might be room for a more complete list of reasons why hyperbolic geometry is important in its own right.

It's hard to know where to start, because there are so many good reasons to study hyperbolic geometry beyond its obvious historical importance in the development of geometry. I'm going to start this list with a few things I can think of off the top of my head; I'll make this answer community wiki so others can elaborate or add to it as they think of other things.

Why is hyperbolic geometry important?

  1. History of math: First, just for completeness, I'll mention the historical reason. The discovery of hyperbolic geometry was, in my opinion, the second most important event in the history of mathematics (the first being Euclid's introduction of the axiomatic method). Up until the nineteenth century, everyone thought of axioms as self-evident truths about the real world, which could be built upon to derive less self-evident truths. Since the discovery of hyperbolic geometry, axioms have been thought of as more or less arbitrary assumptions that could be used to get an axiomatic system started, and then everything proved within that system is exactly as valid as the axioms themselves. It's the foundation of the way we currently understand mathematical truth. (I've written more about this in the preface and Chapter 1 of my undergraduate textbook Axiomatic Geometry.)
  2. Complex analysis: The hyperbolic geometry of the unit disk (or, equivalently, the upper half-plane) plays a central role in complex analysis. For example, the Schwarz-Pick Lemma says that any holomorphic map from the unit disk to itself is either an isometry of the hyperbolic metric or a strict contraction (which decreases all distances). This has important consequences for understanding the nature of holomorphic maps.
  3. Geometry of Surfaces: Most connected surfaces (all but the plane, cylinder, torus, Möbius strip, Klein bottle, sphere, and projective plane) carry a Riemannian metric of constant negative curvature, which is therefore locally isometric to the hyperbolic plane. Moreover, all such surfaces can be realized as quotients of the hyperbolic plane modulo discrete groups of hyperbolic isometries.
  4. Geometry of Three-Manifolds: The geometrization theorem says that every closed 3-manifold can be cut along spheres and tori into finitely many pieces, each of which can be given one of eight possible highly symmetric geometric structures. By far the richest of these structures is hyperbolic geometry, which accounts for "most" three manifolds.
  5. Cosmology: The leading candidates for modeling the shape of the universe as a whole are the FLRW models, in which the spatial geometry of the universe is either flat, spherical, or hyperbolic. Which one depends on the average density of matter and energy and the value of the "cosmological constant."
  6. Fermat's Last Theorem: Wiles's proof of Fermat's last theorem made essential (and unexpected) use of modular forms, which are functions on the hyperbolic plane that satisfy a specific transformation property under a certain discrete group of hyperbolic isometries.
  7. Art: This list wouldn't be complete without mentioning the groundbreaking uses of hyperbolic geometry in the art of M. C. Escher.
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+1) Blue, one needs to look into some history here. That parallel postulate (which was put indeed as an axiom by Euclid) was challenged by many not to be a postulate at all. In fact, many mathematicians and other scientists alike thought that it should have been a theorem rather than just being given as an assumption. When considering it as a theorem, in a nut shell it came down to several cases to be observed when the "proof" was generated. And as it turns out, for many parallel lines through a given point it came a hyperbolic surface and when no parallel line would suit, the surface became a sphere. And so this resulted in many new properties, one of them being that the sum of angles relates to the area of a triangle. Challenging the parallel postulate is just another great example that new doors to unknown worlds open up. A good book (there are several) to read would be : "Introduction to non Euclidean Geometry" by Wolfe (Dover publ).

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    $\begingroup$ I feel I didnt make my question clear. This is precisely the one motivation that I know. But for me, the novelty has worn off and I feel that there should be other ideas that I am just not seeing. $\endgroup$ – blue Aug 26 '15 at 15:52
  • $\begingroup$ @blue I am not aware of any "other" ideas of how this came to be. But I do remember that back in college when I was learning about the background of the parallel postulate and how hyperbolic geometry came to be as a result, I could not sleep that night!!! $\endgroup$ – imranfat Aug 26 '15 at 16:54
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I share the same views of Imranfat.Imagine for close to 19 centuries Euclidean geometry ruled the roost, just a plus minus change at the proper place opened the door to a new world so similar and yet so different and captivatingly beautiful. In the new order there being a teasing familiar part, an unknown part that demands your intellectual attention holding out a hope for new research.

There is not one but a host of novelties out there.The validity and further scope even over and above the three models awaiting new approaches or developments. The refreshingly new definitions of line intersection and parallels that remove a straight line and plug in curved lines in place with fully recognizable validity..All trigonometric functions can be transformed into hyperbolic functions. Triangles with angle excess getting into deficit.

Even its history is interesting.The big Gauss kept it to himself for a long time until its reality had to surface, its discovery and frustrations by Bolyai, Riemann's made distinction between the infinite and the indefinite, Gauss's spontaneous cry of joy after his Dissertation, Beltrami's solid work leading to its firm footing, his paper-mache molded model with saddle points..even today Daina Tamania's crocheting on display at the Smithsonian etc. etc..

Difficult to answer what job opportunities and what extra $ comes out of it, but a concern for any avenue that science takes you through is good. Just wroting in extempore..

Gauss Egregium theorem. You squeeze a hemisphere to see all three type of shapes and even a fourth one a non-symmetrical shape like a huge grain of wheat without rotational symmetry. A curved line on a cone is still straight when opened out flat can be relevant to warped surfaces.. A beautiful mind that sees any surface isometrically embeddable into possible deeper dimension far beyond a geometry what meets the eye, the hyperbolic geometry's unmistable part.

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