This paper is a concise and extremely well-written introduction to hyperbolic geometry. It should be interesting and useful to every mathematician or physicist. The first sections contain an exposition of some important historical points on hyperbolic geometry, from Euclid’s parallel’s axiom until Poincaré, Minkowski and Einstein. Then there is a very cute remark on one-dimensional hyperbolic geometry, and then a generalization to higher dimensions. Then the authors present in detail five models of hyperbolic \(n\)-dimensional space. They describe the most important features (geodesics, isometries, Gauss-Bonnet) and study a sixth model, which is a “combinatorial” model of hyperbolic space. After that they turn to modern hyperbolic geometry and make some very interesting remarks on quasi-conformal maps, Mostow rigidity, Thurston’s work and Gromov-hyperbolic spaces.
The paper contains interesting exercises, bibliography and an index.
For the entire collection see [Zbl 0882.00019].