## Introduction to Hyperbolic Geometry

## Course Advertisement, Summer, 1996

### Math 690, Topics in Mathematics, credits and meeting times to be
arranged

### 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.

For more information about the course contact Dr. Kenneth R. Driessel
via electronic mail at driessel@iastate.edu.

The last update for this file was by Kenneth R. Driessel on June 21,
1996.