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Geometries

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MAA Review

The intent is clear from the very first words of the preface: “This book is dedicated to the proposition that all geometries are created equal.” The pervading idea is to follow the spirit of Felix Klein’s Erlangen program and to regard a geometry simply as set with a transformation group operating on it. It is clear that Klein’s notion does not itself characterize geometries because it is far too broad — even an abstract group meets the criterion. The author’s goal is then to look at interesting and important individual geometries that carry some additional structure, and to look at their relationships within the category of geometries.

For example, to give precise mathematical meaning to Cayley’s famous phrase “projective geometry is all geometry”, the author introduces the idea of a “subgeometry” (an image under an injective equivariant map), and notes that almost all the geometries studied in his book — including the classical hyperbolic, elliptic and Euclidean ones — are subgeometries of projective geometry.

He emphasizes that geometry is a way of thinking about mathematics, and offers an approach that is far more visual than algebraic or analytical. Overall, the book is a kind of tour of geometries and geometric ideas. It is based on lectures that the author gave at the Independent University of Moscow to first year students. In the US it would be accessible to undergraduates, especially those with at least a bit of group theory.

After preliminaries (a little Euclidean geometry, a bit of group theory, and some toy geometries) the author introduces finite subgroups of SO(3) and the Platonic solids followed by geometries based on discrete subgroups of the isometry group of the plane. From there he moves on to Coxeter geometries, then spherical, hyperbolic and projective geometries. (The classical geometries are discussed only in two dimensions.) Each of these gets a rather quick look. The idea is for students to see a broad sampling of geometries and to develop a sense of how they are related.

A final chapter looks at some specific examples of morphisms of geometries. These include the Hopf bundle, the Grassmannian, the Stiefel-over-Grassmann bundle and Milnor’s universal G-bundle. The treatment is short and offers barely of sketch of the underlying ideas but it gives students some notion of a broader picture.

There are a modest number of exercises — averaging ten per chapter — with some solutions and hints. The book would nicely support a seminar or special topics course. It would probably not work well for self-study without additional materials because the treatment of many topics is fairly cursory.

Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

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