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Differential and Complex Geometry: Origins, Abstractions and Embeddings

Raymond O. Wells, Jr.
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Mark Hunacek
, on

Sometimes a book turns out to be much more broadly accessible than its title would suggest that it is. That, I think, is the case with the book now under review. Reading the title of this book, one could be forgiven for thinking it is intended for specialists in the area of differentiable real and complex manifolds. However, the book turns out to be not only a substantive account of some of the mathematics behind differential and complex geometry, but also a very readable slice of the history of geometry over a period of roughly 300 years.

The basic idea behind the book is this: among the mathematical accomplishments of the 20th century was the creation of a theory of differentiable and complex manifolds, followed by a string of embedding theorems that provide conditions under which these objects could be embedded in real or complex \(n\)-dimensional spaces, or projective spaces. The first of these results, the author tells us, was proved by Whitney in 1936 and “ushered in a new era in geometry” by showing that “any differentiable manifold can be embedded as a closed submanifold of a higher-dimensional Euclidean space”.

Of course, most if not all mathematical achievements have a history that can be traced back in time, and these embedding theorems are no exception. The author backtracks hundreds of years and discusses a number of strands of geometry and analysis that come together to produce these results.

More specifically: the book is divided into four parts. Each part begins with a fairly substantial introduction, usually about three to five pages long. These introductions don’t appear in the Table of Contents, which is a pity, since they are helpful in giving a sense of what is to follow.

The first part consists of two chapters. The first is titled “Algebraic Geometry” but may equally well have been called “Analytic and Algebraic Geometry”: after a brief introductory discussion of some ancient Greek work (Appolonius and Pappus), the chapter proceeds to Descartes’ and Fermat’s approaches to analytic geometry, and then ends with Newton’s early 18th-century classification of algebraic curves of degree 3, mentioning briefly Euler’s discussion and analysis of this work. The second chapter in Part I discusses differential geometry, specifically 18th century notions of curvature. The chapter ends with a discussion of some preliminary work of Euler involving the curvature of a surface, work that would be expanded on about 60 years later by Gauss.

This work comprises a portion of Part II of the book, which is captioned “Differential and Projective Geometry in the Nineteenth Century”. It consists of three chapters, the first recounting how the earlier work of people like Desargues was rediscovered in the 19th century, the second and third returning to differential geometry and discussing the work of Gauss and Riemann, respectively.

Part III, consisting of six chapters, is called “Origins of Complex Geometry”, and studies a lot of the history of analysis in the 19th century that led to the theory of complex manifolds. It begins with a chapter on the complex numbers themselves, discussing how Argand, Gauss and Wessel came up, independently, with the idea of representing the complex numbers as points in the plane.

One little quibble: it seemed to me that many students reading this chapter might conclude that the historical motivation for complex numbers was the desire to solve quadratic equations by using the quadratic formula. (“This formula leads to the problem of understanding what one means by the square root in the cases where \(b^2-4ac\) happens to be negative.”) In fact, mathematicians back then were perfectly comfortable with the idea of a quadratic equation having no (real) solutions; what caused problems were cubic equations. These did have real solutions, and sometimes, even when the real solution was obvious by inspection, the formula for the root still led to square roots of negative numbers. So, in this context, people like Bombelli — whose name does not appear in the book — were led to look at complex numbers and their arithmetic.

Additional chapters in Part III of the book sketch the history of elliptic integrals and functions (chapters 7 and 8), a look at the history of complex function theory from the different points of view of Cauchy, Riemann and Weierstrass (chapter 9) and the development of Riemann Surfaces (chapter 10). There is some overlap here between this book and The Real and the Complex by Jeremy Gray; I was surprised to see that the latter book was not listed in the bibliography of the former.

This part of the book then ends with a chapter briefly discussing various 19th century ideas that play an important role in complex geometry. These include transformation groups (now better known as Lie groups), the uniformization theorem for Riemann surfaces, and the rise of algebraic and point set topology, including the abstract definition of a manifold.

Part IV of the book is the destination that everything else has been preparing us for: a look at the embedding theorems of the mid-20th century. The discussion here is somewhat technical, but the author does a good job of summarizing and motivating the basic ideas. Chapter 13, for example, contains a one-section overview of Nash’s embedding theorem (involving isometric embeddings in the context of Riemannian manifolds) followed by several other sections in which the details are fleshed out.

The author’s writing style is generally clear and inviting, and a very nice feature of the book is his use of original sources, pages of which are reproduced throughout the text. The reader will find in chapter 1, for example, a page from Descartes’ La Géométrie, followed by the cover page of Newton’s Opticks.

This is a valuable book. There is sure to be something useful in it for anybody who is interested in geometry, topology, analysis or the history of mathematics.

Mark Hunacek ( teaches mathematics at Iowa State University. 

See the table of contents in the publisher's webpage.