## Introduction to Hyperbolic Geometry

### Prerequisites:

Basic knowledge of standard graduate material in algebra, linear algebra and analysis. In fact, a good undergraduate mathematical background should be more than enough for most of the course. Specifically, Abstract Algebra (Math 301), Topology (Math 331), Theory of Matrices (Math 307), and Advanced Calculus (Math 414 or 465) or consent of the instructor.

### Course description:

Euclid's parallel postulate is equivalent to the following statement: Through a point outside a straight line there is a unique straight line passing through the point that does not meet the given line. In hyperbolic geometry there is more than one such parallel line. Early in the 19th century, Bolya, Lobachevsky and Gauss developed hyperbolic geometry using traditional synthetic procedures. In 1868 Beltrami used differential geometry to present a Euclidean model of hyperbolic geometry; he proved that hyperbolic geometry is at least as consistent as Euclidean geometry. In 1872, Klein suggested that geometries should be studied by the analysis of their isometry groups.

In this course, we shall roughly follow this historical outline. First, we shall develop hyperbolic geometry from a modern set of axioms. After an introduction to differential geometry, we shall prove the relative consistency of these axioms. We shall then study the groups of isometries of the hyperbolic plane and hyperbolic three space. We shall see, for example, that the isometry group for the hyperbolic plane is closely related to the group SL(2,R) of 2 by 2 real matrices with determinant equal to 1. Finally, we shall study the relationship between these isometry groups and the groups associated with the theory of special relativity (which are call Lorentz groups).

We shall study the book `Introduction to Hyperbolic Geometry' by Arlan Ramsay and Robert Richtmyer, published by Springer in 1994.

Reference: J. G. Ratcliffe. `Review of Introduction to Hyperbolic Geometry' by Ramsay and Richtmyer', Am. Math. Monthly, 1996, p. 185 ff. You may also look at a list of further references.