Geometry With An Introduction To Cosmic Topology

Geometry with an Introduction to Cosmic Topology is motivated by questions that have ignited the imagination of stargazers since antiquity. What is the shape of the universe? Does the universe have an edge? Is it infinitely big? Dr. Hitchman aims to clarify this fascinating area of mathematics and focuses on the mathematical tools used to investigate the shape of the universe. The text follows the Erlangen Program, which develops geometry in terms of a space and a group of transformations of that space. This approach to non-Euclidean geometry provides excellent material by which students can learn the more sophisticated modes of thinking necessary in upper-division mathematics courses. This unique text is organized into three natural parts: Chapter 1 introduces the geometric perspective taken in the text and the motivation for the material that comes from cosmology. Chapters 2-7 contain the core mathematical content of the text, developing hyperbolic elliptic, and Euclidean geometry from the complex plane and subgroups of Mobius transformations. Other topics include the topology and geometry of surfaces and Dirichlet domains. Finally, Chapter 8 explores the topic of cosmic topology through the geometry learned in the preceding chapters.