Prior to looking at this book, I had always assumed that there was a fairly clear-cut distinction between undergraduate and graduate differential geometry. The former subject — once a fairly standard course offering in the American undergraduate curriculum, now unfortunately much less often seen by undergraduates — involved curves and surfaces in the Euclidean plane and three-dimensional space (as in, for example, *Elements of Differential Geometry* by Millman and Parker, or, more recently, *Differential Geometry of Curves and Surfaces* by Banchoff and Lovett), and the latter studied abstract differentiable manifolds, typically with the ultimate goal of defining a Riemannian metric on the manifold and thus being able to study curvature. (See, e.g., a trilogy of excellent books by John Lee: *Introduction to Topological Manifolds*, *Introduction to Smooth Manifolds *and finally *Riemannian Manifolds: An Introduction to Curvature*.)

The book under review intentionally blurs this distinction and also introduces topics (like contact geometry and symplectic geometry) that are sometimes not even encountered in introductory graduate courses, much less undergraduate ones. It therefore presents a view of differential geometry that is somewhat more sophisticated than the coverage provided in the two undergraduate books cited above.

One of the reasons that I asked to review this book (which, being in Springer’s *Undergraduate Texts in Mathematics* series, is clearly intended for an undergraduate audience), was to see if the material mentioned in the title could even *be* reasonably presented to undergraduates. The book convinces me that it can. The author’s description of the intended audience is “students who have completed the traditional calculus and linear algebra sequence but who have not necessarily been exposed to the more abstract formulations of pure mathematics.” That strikes me as somewhat optimistic; to me, the book seemed more appropriate for a course at the junior — or, more likely, senior — level. Although the author takes pains to adhere to the modest prerequisites cited in the preface, I can’t help but think that considerable mathematical maturity, beyond that possessed by most sophomores, would also be necessary.

The author does make a considerable effort to keep things as accessible as possible, with fairly detailed explanations, extensive motivational discussions and homework problems that emphasize examples rather than deep proofs, but there is simply no denying that the subject matter of this text (differential forms, symplectic geometry, etc.) is fairly sophisticated and abstract, and not the usual fare for young undergraduates fresh out of calculus and linear algebra.

The first three chapters, comprising about a third of the text, are prefatory to everything that follows, although there is some material here that will quite likely be new to students. The first (rather short) chapter talks about the basics of sets and functions. The second is on linear algebra, and is intended, for the most part, to be a review of material that the student has seen before in a linear algebra course, although some topics here (e.g., the dual space and linear symplectic forms) are rarely if ever taught in such courses. Recognizing this, the author increases the level of detail when discussing these topics. The third chapter is what I would call “advanced calculus done right”; i.e., advanced calculus from the standpoint of linear mappings as derivatives, with the ultimate goal being the introduction of the various objects of study — “geometric sets”, which are essentially submanifolds of \(\mathbb{R}^n\), and tangent spaces defined at each point of them. The author gradually develops an analytic definition of tangent space by first appealing to the reader’s geometric intuition and then carefully expanding on this definition. It is seen that tangent spaces are linear approximations to (typically nonlinear) geometric sets, just as derivatives are linear approximations to (typically nonlinear) functions, and that derivatives are linear mappings from one tangent space to another. The treatment of all of this is careful and consistently well-motivated.

The next chapter, on differential forms and tensors, is motivated by a desire to discuss measurement of objects. Differential forms are defined via the previously-developed linear algebra, as certain kinds of multilinear *k*-forms; the chapter proceeds to discuss the basic operations on them, including integration (Stokes’ theorem is stated in its general form, but not proved; references are provided instead), and then goes on to discuss tensors and the Lie derivative.

With all this (fairly technical) machinery developed, the author is ready to plunge into the three geometries of the book’s title: Riemannian, contact and symplectic, which are discussed in that order in the last three chapters of the book. Riemannian geometry, the subject of chapter 5 of the text, is, of course, the one most commonly taught (especially at the undergraduate level); in fact, as the author notes, many books treat Riemannian geometry as being *synonymous* with differential geometry, rather than as being one of several possible differential geometric structures one might study.

In fact, some undergraduate texts, including the one by Banchoff and Lovett referred to in the first paragraph above, don’t even use the phrase “Riemannian geometry” at all, or even discuss differential forms, but the basic ideas discussed in these books, such as curvature and geodesics, are ideas that can be generalized to Riemannian structures; the book by Millman and Parker cited earlier illustrates this by first discussing these ideas in a very concrete, classical way (without differential forms), and then ending with a chapter introducing the notion of a manifold and Riemannian structure. Chapter 5 of this book can be viewed as a fairly high-level approach to these classical topics, and certainly a more modern approach than the one taken in some of the more elementary texts.

And that brings us to the last two chapters of the book, on contact and symplectic geometry, two geometries that are somewhat akin to one another, although contact geometry lives in odd-dimensional spaces while symplectic geometry lives in even-dimensional ones. I must confess that before looking at this book, I had never even *heard* of contact geometry (that’s *another* reason I asked to review it), but my embarrassment about this is ameliorated to some extent by the fact, pointed out by the author, that the first graduate-level text on contact geometry was not even published until 2008. The author discusses this first in three-dimensional space, and extends the ideas to arbitrary odd-dimensional spaces in a section at the end of the chapter. Before getting to that point, however, there are two sections of motivational material, using Huygens’s Principle in physics and the theory of characteristic curves for differential equations to illustrate ideas that will be generalized in the rest of the chapter.

The basic idea stems from the fact that it is often fruitful to think of familiar objects (such as the set of ordered triples comprising Euclidean three-space) in different ways: for example, an ordered triple \((x, y, m)\) can be identified with a pair consisting of a point \(P = (x,y)\) in the plane and a line \(\ell\) with slope \(m\) passing through \(P\); we call such a point-line pair \((P,\ell)\), where \(\ell\) is a non-vertical line through \(P\), a “contact element”. The set of contact elements can thus be identified *as a set* with three-space, but we visualize contact elements as being something other than points in space. On this set one can construct, using a nondegenerate one-form, a differential geometric structure; this is the beginning of contact geometry. The rest of the chapter pursues these ideas in some detail.

Contact geometry, as noted previously, has a close connection with symplectic geometry, and in the last chapter of this book the latter geometry is discussed, with its connections to contact geometry emphasized. As in the previous chapter, this one begins with a motivational section; historically, the main motivation for symplectic geometry was the theory of Hamiltonian mechanics, and the author briefly surveys this area of physics and explains how symplectic geometry provides a good formulation of it.

The rest of the chapter discusses the basic ideas of the theory in more detail. First, symplectic spaces are defined (just as Euclidean geometry dealt with nondegenerate positive definite forms, here we deal with symplectic forms that are skew-symmetric); then, the basic concepts are dealt with, including a discussion of diffeomorphisms of, and certain geometric sets in, symplectic spaces.

It should be clear from the foregoing summary that there is a lot of mathematics covered in this book. Unless the students are well-prepared and some of the early material can be skipped or covered lightly, this book should take about two semesters to cover. However, the text does seem flexible, and the last three chapters appear to be independent of one another, so one approach to keeping things within the confines of a one-semester course would be to select one of the three geometries of the title and concentrate on that. In particular, a modernized version of the basic undergraduate differential geometry course could emphasize just the first five chapters.

In summary: as previously noted, this book provides a different way of looking at the subject of differential geometry, one that is more modern and sophisticated than is provided by many of the standard undergraduate texts and which will certainly do a good job of preparing the student for additional work in this area down the road. With the right group of students, this book could be the basis of a thought-provoking and interesting class. However, you will definitely need the right group of students.

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