[Math] Motivations for Hyperbolic Geometry

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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?

Best Answer

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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