An Algebraic Approach to Geometry: Geometric Trilogy II

As anybody who has ever taught an introductory linear algebra course knows, many of the ideas presented there can (and should) be geometrically motivated. As an undergraduate, I learned from David Bloom (who went on to discuss these ideas in his book Linear Algebra and Geometry, now out of print) that the process can be reversed: using linear algebra, one can define geometric concepts, and thus give precise algebraic proofs to geometric theorems. I was lucky enough to have a senior seminar based on Snapper and Troyer’s Metric Affine Geometry, a book that explored affine geometry from the standpoint of linear algebra and then discussed the geometric significance of bilinear forms on such spaces. Unfortunately, neither of these two books covered projective geometry in any real detail, but after graduating from college I learned from Kaplansky’s Linear Algebra and Geometry how projective spaces can be defined from linear algebra. More recently, Gallier’s Geometric Methods and Applications for Computer Science and Engineering also pursues these ideas, and quite a few others as well.

And now we have another excellent book to add to the mix. The book under review is the second of a trilogy of books discussing the broad sweep of geometry and the various ways of approaching this subject. The first book of the series, An Axiomatic Approach to Geometry discusses geometry from an historical, axiomatic approach, starting with the pre-history of the subject and proceeding through the contributions of the ancient Greeks through more recent developments, ultimately arriving at the foundations of a rigorous axiomatic approach and the logical consistency and basic theorems of hyperbolic geometry. In the third book in the series, A Differential Approach to Geometry, the author shows how the use of calculus sheds light on geometric ideas. In this book, however, the focus is on algebra, both abstract and linear. A good first course in each subject should be viewed as a minimal prerequisite for the book, though a series of appendices discusses a number of topics in algebra that a student might not be familiar with (for example, homogenous polynomials, symmetric polynomials, resultants, quadratic forms and dual spaces).

The book begins with a chapter on classical analytic geometry, discussing the work of Fermat and Descartes, and the theory of conics and quadrics. Some attempt is also made in this chapter to motivate the more abstract, algebraic definitions that will follow. Chapters 2 through 6, the heart of the book, proceed to do what was described above: define various geometries (affine, Euclidean, projective) in terms of linear algebra, and use that algebra to prove geometric facts. Affine geometry is treated in chapters 2 and 3. Very loosely speaking, affine geometry is Euclidean geometry without distance or measure; affine results are therefore the ones that depend only on concepts such as incidence and parallelism. Interestingly, though, one can, in affine geometry, speak of the ratio of the lengths of two parallel line segments, so that even though distance is undefined, it does make sense to speak of a line segment AB being half the length of a parallel line segment CD. So, for example, the theorem “the line segment joining the midpoints of two sides of a triangle is parallel to and one-half the length of the third side” is a theorem in affine geometry, but the Pythagorean Theorem is not.

The linear-algebraic formulation of an affine space typically involves a vector space V operating on a set E, subject to certain axioms. For a brief discussion of this process, see my review, in this column, of Tarrida’s Affine Maps, Euclidean Motions and Quadrics. The basic idea is that any two points P and Q in the set E give rise to a vector \(\overrightarrow{PQ}\) in V, which we think of as an arrow or line segment drawn from P to Q; this vector acts on P by moving it to Q. Chapter 2 of the book under review discusses all this, proves some results in affine geometry, and analyzes affine transformations. (Because an affine space involves two sets E and V, the very definition of an affine transformation requires some care, but the author explains all this quite clearly.)

Then, in chapter 3, this program is continued, with the underlying field now being the field of real numbers. One significance of this restriction is that it now makes sense to speak of positive and negative field elements; this allows us to speak of orientation in our affine space, as well as the half-plane determined by a line. It also allows the introduction of a notion of “betweenness” into our affine geometry that does not exist when the field is arbitrary, and this in turn allows us to discuss convexity. Another significance of the restriction is that all nonnegative real numbers have square roots, which allows a sharper analysis of quadrics.

After these two chapters, the book turns from affine to Euclidean geometry. This transition also requires that we work over the field of real numbers, and in addition requires the introduction of a positive definite inner product into our vector space V. This is the subject of chapter 4, in which various theorems related to length, distance and angle are discussed. In chapter 5, a short chapter, the author describes what modifications are necessary to pass from real inner product spaces to complex ones. When one works over either the field of real or complex numbers, the classification of quadrics can be described in more detail than when the field is completely arbitrary, and the author discusses this at some length.

After three chapters of working with the fields of real or complex numbers, chapter 6 returns to arbitrary fields and the introduction of projective geometry. Here, the “points” of the geometry are simply the one-dimensional subspaces of some vector space V. (This makes clear how one tags a point with homogenous coordinates — just work with an ordered basis.) The author discusses some major geometric results connected with these projective spaces induced by vector spaces, including the theorems of Desargues and Pappus. He then discusses the axioms for projective planes, which invites comparison to the chapter on projective geometry in Axiomatic Approach, the first volume of the trilogy. Results from that book are used here, without reproving them, to identify which of the axiomatically defined projective planes are those induced by vector spaces. The short and beautiful answer: a projective plane is induced by a vector space over a field if and only if Pappus’ theorem holds; it is induced by a vector space over a division ring if and only if Desargues’ theorem holds. Conics in the projective plane are discussed at some length, along with their n-dimensional analog, projective quadrics. Finally, the chapter ends with a discussion of some topics where the vector space is over the field of real numbers, including a brief look at the topology of the real projective plane.

I thought these five chapters were very successful. The writing was clear, there were lots of examples and exercises, and the discussions were quite well motivated. What I also particularly liked was the balance drawn by the author between the algebra and geometry. Neither overwhelmed the other, and the symbiotic relationship between the two was always apparent. One of my criticisms, for example, of Linear Algebra and Geometry by Shafarevich and Remizov was that the geometry was lost behind the algebra (the chapter on projective geometry there did not even mention the theorems of Desargues and Pappus), but that certainly is not the case here; there are lots of geometric theorems, and helpful diagrams, in these chapters.

The book, however, has more to offer; one remaining chapter provides a brief introduction to algebraic geometry. In an effort to keep things reasonably concrete and accessible, the author restricts attention to algebraic curves (defined by a single polynomial) in complex projective plane. This restriction still allows for a lot of interesting mathematics to be discussed; after all, entire books have been written about algebraic curves. (A nice example is A Guide to Plane Algebraic Curves by Kendig.) In this chapter, we get a discussion of, among other things, singularities, Bezout’s theorem, and the group law on a cubic. Obviously, since this is only a chapter of a book, the coverage here cannot be as thorough as it would be in a full-length book; the Riemann-Roch theorem, for example, is not discussed. But there is plenty here to whet the appetite of potential algebraic geometers.

Most universities these days have neither the money nor manpower (nor, to be sure, the desire) to offer courses along the lines of this book; the material, though interesting and valuable, is really somewhat outside the mainstream of what an undergraduate mathematics major really needs to learn in the two years or so in which he or she has access to upper-level courses. As a result, this book will probably not find much use as an actual text, though I think a senior seminar along the lines of the one I had with Snapper and Troyer would be a great idea. This book would be an excellent candidate for such a seminar, and would give the students not only a new look at geometry but also a renewed appreciation for the power of linear algebra in other branches of mathematics. In any event, regardless of how frequently this book might get used as a text, it is certainly something that any instructor interested in geometry or algebra would want to have. I haven’t read the third book in the Borceux trilogy yet, but based on the first two entries, it is shaping up to be an exceptional piece of work.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.