Getting five books for the price of one is always a pleasant experience, particularly when all five books are excellent. That is, I am happy to report, precisely the case with the book under review — which, despite being in Springer’s “Texts in Applied Mathematics” series and having the word “Applications” in the title, should also be of considerable interest to pure mathematicians with no interest in applications at all. The author writes in his preface that this book is an “introduction to fundamental geometric concepts and tools needed for solving problems of a geometric nature with a computer”; these concepts cover a wide range, and, as a result, anyone who likes to read about geometry, differential geometry, linear algebra, or Lie theory should find something of interest in this book.
First and foremost, this is, of course, a text in geometry, and many aspects of it are covered. After a brief but interesting introductory chapter which rapidly surveys the history of geometry and discusses various kinds of geometries, chapters 2, 4, 5 and 6 provide a basic course in affine, Euclidean and projective geometry, emphasizing linear algebra throughout. (A solid undergraduate course in linear algebra is a prerequisite for the entire text, and a few chapters require some understanding of analysis as well.)
Affine geometry is defined in terms of a vector space acting on a nonempty set in such a way as to satisfy certain axioms. Given two elements a and b in our “object space”, the vector which acts on a to produce b is denoted . Intuitively, we think of this vector as an arrow starting at a and ending at b. Within this framework one can discuss lines, planes, and their higher-dimensional analogues, as well as geometric notions involving incidence and parallelism. Although one can compare, in affine geometry, the ratio of parallel segments, there is no notion of distance or angle; the proper tool for this is an inner product. Consequently, the chapter on Euclidean geometry is essentially a chapter on real inner product spaces and orthogonal mappings and matrices. The chapter on projective spaces also gives a linear algebraic, rather than axiomatic, development: the points of a projective k-space are defined as the one-dimensional spaces of a (k+1)-vector space, etc. This has the effect of limiting the projective spaces that are considered (for example, all projective planes of this sort satisfy both the theorems of Desargues and Pappus, which is not the case with all projective planes defined by the standard axioms), but these spaces are still general enough to allow discussion of many topics in the subject: cross ratio, duality, projective mappings, etc.
I should, in the interest of full disclosure, confess to a pre-existing bias here. Ever since college, when I attended lectures by David Bloom on linear algebra and geometry (which inspired a textbook by him on this subject which is, sadly, out of print; where is Dover Press when you need it?), I have been very interested in the interplay between these two subjects, to the point where I contributed a chapter to Hogben’s Handbook of Linear Algebra that is somewhat along the lines of chapters 2, 4, 5 and 6 in this textbook, though covering less material and with all proofs omitted. In graduate school, my interest in this area grew to include geometric transformations and matrix groups, and of course from there it is not a big jump to Lie theory. All of these topics are also, as will be pointed out shortly, covered in this book, so in some sense I was predisposed to find it interesting, but what really sold me on this book was the beautiful exposition. Gallier has an enviable ability to provide intuitive motivation without sacrificing mathematical rigor; I enjoyed, for example, his frequent use of the Bourbaki “curves ahead” symbol.
Several other chapters in this book (3, 7, 8, 9 and 10) provide enough material to comprise a second, or topics, course in Euclidean geometry. Chapters 3 and 7 discuss convexity — the first of these discussing the basics of the subject (the highlights being the major theorems of Cartheodory, Radon and Helly), the second discussing separating hyperplanes and the geometric form of the Hahn-Banach theorem (in finite-dimensional spaces). Chapter 8 provides a nice look at geometric transformations, including the Cartan-Dieudonne theorem, and this subject is further explored in chapter 9, which discusses rotations of R3 by means of quaternions, an approach which also allows a discussion of rotations in four-dimensional space and a look at the double covering from SU(2) to SO(3). (Another way in which quaternions arise in the study of matrix Lie groups is that, just as the special orthogonal group SO(n) can be viewed as the group of rotations of n-dimensional real space and the special unitary group SU(n) can be viewed as the group of rotations of n-dimensional complex space, the symplectic group Sp(n) can be viewed as the group of rotations of n-dimensional quaternion space. This approach, though not developed in this book, is discussed in Stillwell’s Naive Lie Theory and Tapp’s Matrix Groups for Undergraduates.)
The final chapter listed above, chapter 10, discusses some topics in computational geometry, specifically Voronoi diagrams and Delaunay triangulations, neither of which, I confess, I had ever heard of before seeing them here. The presentation is, intentionally, somewhat sketchy and many results are stated without proof, but people interested in topics in Euclidean geometry might enjoy being exposed to these concepts.
Two chapters in this book (19 and 20) provide an introduction to differential geometry of curves and surfaces, respectively; the author states that most of chapter 20 is based on upper-division undergraduate lectures given by Calabi in 1994. These two chapters give, in about 125 pages, a beautifully concise introduction to the subject, even stating and discussing (but not proving) the Gauss-Bonnet theorem. Manifolds are mentioned but not discussed in depth.
In addition to all this geometry, the book also has enough material to cover a course in advanced topics in linear algebra (though naturally those topics that are relevant to geometry are stressed). Complex inner product spaces and spectral theory are covered in chapters 11 and 12; the singular value decomposition, polar form, pseudo-inverses and applications of these ideas are discussed in chapters 13 and 14; chapter 15 is about quadratic optimization; chapter 16 addresses the subject of Schur complements. Recent work on applications of linear algebra to contour grouping in computer vision is the subject of chapter 17, but I must again confess that, being totally ignorant in this area, I didn’t really get as much out of this chapter as people with some background in the area might. However, chapters 11 through 16, by themselves, comprise an excellent second course in linear algebra. (However, one topic that an instructor may want to cover in such a course — canonical forms — is not developed in the book, but the existence of the Jordan form is assumed in at least one exercise later on. Speaking of exercises, there are a plentiful supply of them, most of which struck me as non-routine.)
As if all the preceding were not enough, the book also contains a nice introduction to matrix Lie groups and their Lie algebras. Chapter 9, as previously noted, introduced some of these ideas, and that chapter can be viewed as leading directly to chapter 18, which pursues matters further by using the matrix exponential to provide a bridge between classical real and complex matrix groups (general linear, special linear, orthogonal and unitary and their special counterparts, and the Lie algebras associated with them).
In addition to almost 700 pages of text, there is an accompanying website which contains yet more text-quality material; the URL of this webpage, mentioned in the preface, is http://www.cis.upenn.edu/~jean/gbooks/geom2.html. Although originally written to accompany the first edition of this book, there is a lot of material here that is of interest to the second edition as well.
While it should be apparent from what has been said to this point, let me make it very explicit: this book is filled with interesting mathematics, superbly presented. Aside from its potential use as a text, the book should be looked at by anyone who uses or is interested in the topics covered. It is highly recommended.
Mark Hunacek (firstname.lastname@example.org) teaches mathematics at Iowa State University.