Geometry with an Introduction to Cosmic Topology

Preface Preface

Geometry with an Introduction to Cosmic Topology approaches geometry through the lens of 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?

This text develops non-Euclidean geometry and geometry on surfaces at a level appropriate for undergraduate students who have completed a multivariable calculus course and are ready for a course in which to practice the habits of thought needed in advanced courses of the undergraduate mathematics curriculum. The text is also suited to independent study, with essays and discussions throughout.

Mathematicians and cosmologists have expended considerable amounts of effort investigating the shape of the universe, and this field of research is called cosmic topology. Geometry plays a fundamental role in this research. Under basic assumptions about the nature of space, there is a simple relationship between the geometry of the universe and its shape, and there are just three possibilities for the type of geometry: hyperbolic geometry, elliptic geometry, and Euclidean geometry. These are the geometries we study in this text.

Chapters 2 through 7 contain the core mathematical content. The text follows the Erlangen Program, which develops geometry in terms of a space and a group of transformations of that space. Chapter 2 focuses on the complex plane, the space on which we build two-dimensional geometry. Chapter 3 details transformations of the plane, including Möbius transformations. This chapter marks the heart of the text, and the inversions in Section 3.2 mark the heart of the chapter. All non-Euclidean transformations in the text are built from inversions. We formally define geometry in Chapter 4, and pursue hyperbolic and elliptc geometry in Chapters 5 and 6, respectively. Chapter 7 begins by extending these geometries to different curvature scales. Section 7.4 presents a unified family of geometries on all curvature scales, emphasizing key results common to them all. Section 7.5 provides an informal development of the topology of surfaces, and Section 7.6 relates the topology of surfaces to geometry, culminating with the Gauss-Bonnet formula. Section 7.7 discusses quotient spaces, and presents an important tool of cosmic topology, the Dirichlet domain.

Two longer essays bookend the core content. Chapter 1 introduces the geometric perspective taken in this text. In my experience it is very helpful to spend time discussing this content in class. The Coneland and Saddleland activities (Example 1.3.5 and Example 1.3.7) have proven particularly helpful for motivating the content of the text. In Chapter 8, after having developed two-dimensional non-Euclidean geometry and the topology of surfaces, we glance meaningfully at the present state of research in cosmic topology. Section 8.1 offers a brief survey of three-dimensional geometry and 3-manifolds, which provide possible shapes of the universe. Sections 8.2 and 8.3 present two research programs in cosmic topology: cosmic crystallography and circles-in-the-sky. Measurements taken and analyzed over the last twenty years have greatly altered the way many cosmologists view the universe, and the text ends with a discussion of our present understanding of the state of the universe.

Compass and ruler constructions play a visible role in the text, primarily because inversions are emphasized as the basic building blocks of transformations. Constructions are used in some proofs (such as the Fundamental Theorem of Möbius Transformations) and as a guide to definitions (such as the arc-length differential in the hyperbolic plane). We encourage readers to practice constructions as they read along, either with compass and ruler on paper, or with software such as The Geometer's Sketchpad or Geogebra. Some Geometer's Sketchpad templates and activites related to the text can be found at the text's website.