Geometry with an Introduction to Cosmic Topology

Geometry with an Introduction to Cosmic Topology, written by Michael Hitchman, is a masterfully written textbook, notable for its clarity of exposition, balance of breadth and depth, and exceptionally motivating theme… after all, what could be more compelling than studying fundamental properties of the universe?

Hitchman describes Cosmic Topology as “the effort to determine the shape of our universe through observational techniques.” The geometry of our universe may be Euclidean, Hyperbolic, or Elliptic. To present a unified framework for these various geometries, the author embraces Felix Klein's “Erlangen Program.” This approach to geometry begins with a space and a group of transformations; it is determined what properties are preserved by these transformations, and these properties (for example, length, angle, incidence, or cross-ratio) become the objects of study. For students, this may initially seem “backwards” compared to Euclid's approach to geometry, which begins by defining the objects of study and the axioms they satisfy. However, through the course of this text, it becomes apparent that Klein's program yields great dividends.

Naturally, the text begins with complex numbers, providing a thorough treatment of complex arithmetic, standard form (a + bi) and polar form (re^iθ), and complex equations for lines and circles. Following this is a chapter discussing complex transformations: general linear transformations (which include translations, rotations, and dilations), reflection across lines, inversion in circles, and Möbius transformations. The reader will find a balanced treatment of computational and theoretical material, and the author stays true to the theme of the Erlangen Program (for example, proving that rotations and translations preserve angles and Euclidean distance).

Readers are gently introduced to Klein's approach to geometry by simple examples of geometries defined by a group of transformations: translational geometry, rotational geometry, Euclidean geometry, and Möbius geometry. The author details the relationship between the objects of study and the transformation groups in each case.

Hyperbolic geometry is treated next, primarily using Poincaré's unit disk model. A great number of figures help the reader visualize how these transformations move various points within the hyperbolic plane. The formula for measuring distances in hyperbolic geometry is masterfully developed. Also included are hyperbolic versions of the law of cosines, the law of sines, and the Pythagorean theorem (whose proofs are outlined in a series of exercises). Afterwards, the author moves on to develop integral formulas for calculating arc length and area; for these sections, the reader should be comfortable with multivariable calculus. The venture into hyperbolic geometry concludes with an optional section on the area of a hyperbolic triangle. To simplify the computations in the derivation of this formula, the author considers a second model for hyperbolic geometry — the upper half-plane model — and so he includes formulas for transformations between these two models, enabling us to apply results from previous sections.

The following topic is elliptic geometry, introduced via geometry on the sphere that has been stereographically projected onto the extended complex plane. Antipodal points are identified, so that between every pair of points there will be a unique line; the result is the disk model for elliptic geometry. The topics covered parallel those from hyperbolic geometry: distances between points, arc length, surface area, and trigonometric formulas.

Next, the author devotes a chapter to curvature and explains how changing the radius of the disk used to model hyperbolic or elliptic geometry will change the formulas developed in previous chapters (in which the radius of the disk equals one). Following this is a whirlwind tour of fundamental and popular topics from topology, such as homeomorphisms, surfaces, orientability, Euler characteristic, and quotient spaces. These topics are brought together in a discussion of which geometries various topological surfaces admit. The text concludes grandly with a discussion of two programs in cosmic topology, the cosmic crystallography method and the circle-in-the-sky method, either of which may someday yield information as to the shape of the universe.

In summary, Geometry with an Introduction to Cosmic Topology presents a high-quality, accessible exposition of Klein's transformational approach to geometry, ideal for an audience of junior or senior level mathematics students. There is a great selection of exercises of varying difficulty. The theme of “cosmic topology” provides both focus and a compelling motivation. Altogether, this is an exciting and innovative geometry text.

Lee Stemkoski is Assistant Professor of Mathematics at Adelphi University.