Abstract
This chapter is concerned with hyperbolic (Lobachevskian) geometry, which is tightly linked to linear algebra through a model in which hyperbolic space is represented as the projectivization of the interior of the light cone in the corresponding pseudo-Euclidean space. The chapter begins with a detailed description of this model and a study of hyperbolic geometry. After that, the axiomatic foundations of hyperbolic geometry are discussed, and a brief historical overview is given. At the end of the chapter, some geometric notions (distance, angle, etc.) are introduced, and basic facts of hyperbolic geometry are established. A brief discussion of elliptic geometry is also presented.
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Notes
- 1.
Here we shall follow the ideas of Boris Nikolaevich Delaunay, or Delone (1890–1980), in his pamphlet Elementary Proof of the Consistency of Hyperbolic Geometry, 1956.
- 2.
Some of these are proved in first courses in geometry, and in any case, elementary proofs of all of these results can be found in Chap. 2 of the book Higher Geometry, by N.V. Efimov (Mir, 1953).
- 3.
For example, his first published book, Disquisitiones Arithmeticae, was considered for a long time to be quite inaccessible.
- 4.
More about this can be found, for example, in the book A Course of Differential Geometry and Topology, by A. Mishchenko and A. Fomenko (Mir, 1988).
- 5.
Their proofs are given in every course in higher geometry, for example, in the book Higher Geometry, by N.V. Efimov, mentioned earlier.
- 6.
Of course, here we are assuming that they all satisfy the axioms of groups I–IV.
- 7.
Felix Klein. Nicht-Euklidische Geometrie, Göttingen, 1893. Reprinted by AMS Chelsea, 2000.
- 8.
Elliptic geometry is sometimes called Riemannian geometry, but that term is usually reserved for the branch of differential geometry that studies Riemannian manifolds.
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© 2012 Springer-Verlag Berlin Heidelberg
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Shafarevich, I.R., Remizov, A.O. (2012). Hyperbolic Geometry. In: Linear Algebra and Geometry. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30994-6_12
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DOI: https://doi.org/10.1007/978-3-642-30994-6_12
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