Lectures on Differential Geometry

I’m going to begin this review by saying something completely ridiculous that is nevertheless true: I love Russian mathematics textbooks. I have a profound fascination for the works of Russian mathematicians, particularly those by the faculty or students of the legendary Mechanics and Mathematics Department at Moscow State University. The names affiliated with this Mecca of scientific study are legendary. Komologrov, Gelfand, Naimark, Petroskii, Landau, Fedeev, Postnikov, Shafarevich, Vinberg, Novikov, Arnold, Zorich — just to name a few. But even more significant is the overriding philosophy of the department, best exemplified by the their very name: Mechanics and Mathematics. The Russian school has traditionally believed in the inseparability of the mathematical and empirical sciences, an approach summed up by the words of Arnold: “Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap.”

Having been trained in biochemistry with heavy leanings towards physical chemistry before being seduced into pure mathematics, I have a deep sympathy for this viewpoint. Cut off from the natural sciences, mathematics seems to become merely an exercise in logic splitting and diagram chasing. I think Lebesgue stated my opinion on this matter best: “Reduced to general theories, mathematics would become a beautiful form without content. It would quickly die.” The partial reunification of theoretical physics and pure mathematics brought about in the last thirty years, and the resulting explosive developments in operator theory, noncommutative geometry, quantum algebra and deformation theory, have underscored Lebesgue’s assessment.

Which brings me in a roundabout way to the blue paperback before me titled Lectures On Differential Geometry by Iskander A. Taimanov. The author’s name should be familiar — a doctoral student of Novikov, he has published many new results on dynamical systems theory. His name may also be familiar to students, in that he has co-authored two popular textbooks that have been translated into English and published by the AMS: Geometry, coauthored with the eminent topologist V. Prasolov and the more advanced and awesome tome, Modern Geometric Structures and Fields, coauthored with his mentor, S. P. Novikov.

MGSF has already been reviewed here and there’s no need to dwell on its level of excellence, but I will say that there is probably no text that better exemplifies the unity approach above better then this one. Any library of a practicing mathematician or physicist that does not include a copy will be very much poorer for it. One cannot help but compare the two books, despite the more modest aims of Taimanov’s solo book. As expected, there is quite a bit of overlap, but this book stands very well on its own.

Differential geometry has always been one of my favorite subjects. It always seemed to me to be an incredibly intuitive subject — especially the classical version in Euclidean space. I first studied classical differential geometry out of Do Carmo’s Differential Geometry of Curves and Surfaces and the 2^nd edition of O’Neill’s Elementary Differential Geometry. Armed with basic calculus, the first and second fundamental forms and linear algebra in your toolbox, it seems if you’re clever enough, you can prove just about anything about any geometric structure in Rⁿ. I remember proving for a homework problem set that if a surface S in R³ consists entirely of umbilic points (i.e. the normal curvatures of all the unit tangent vectors of any point on S are constant and identical), then S is either part of a plane or part of a sphere in R³.I thought it was the coolest magic trick I’d ever seen.

Some of the magic is lost when one moves from Euclidean space to smooth manifolds, where things cannot be so easily visualized, but the idea remains the same: Calculus and (multi)linear algebra are used to completely describe geometric objects in topological spaces. Another “Russian” style text on this subject — and probably one with the deepest ties to physics — is a cause for excitement for me.

This book developed from Taimanov’s undergraduate lecture course at Novosibirsk State University on differential geometry. A glance at the contents demonstrates how inferior most of our mathematics undergraduates would be compared to those at a good Russian university! The first two chapters develop classical differential geometry i.e. the geometry of curves and surfaces in R^n.

Chapter 1 covers the basics of curves in Euclidean space. It’s in the last part of chapter 1 that the book starts to deviate from conventional presentations. Firstly, it gives a very complete and simple discussion of the orthogonal group of motions in Rⁿ. This demonstrates one of the fundamental differences between the training in Russian universities and in the U.S.: an early and very fundamental emphasis on abstract algebra. Students are generally expected to have mastered a very large part of linear algebra (up to and including bilinear forms) as well as the elements of group theory well before taking any advanced courses. Insisting on this early and deep training in algebra is critical to the geometric presentation in this book, which is simultaneously quite concrete and yet totally rigorous. It is a lesson all of us could take under advisement when planning departmental curricula.

Also at the end of chapter 1 is the definition of n-dimensional smooth submanifold in R^n+m. Several “Western” introductions to differential geometry do use a similar approach (O’Neill, for example) — but this is usually much later in the text, after a great many classical cases have been studied in detail. I’m not sure which approach I’m most comfortable with, but it certainly indicates how much preparation Taimanov expects of his students.

Chapter 2 presents the elements of surface theory in Rⁿ. Again, this chapter covers all the usual suspects: the first and second fundamental forms, curvatures of curves embedded in surfaces, Gaussian curvature and principal directions, normal curvature, a very geometric discussion of the covariant derivative, geodesics and an unusually detailed derivation and proof of the Gauss-Bonnett theorem. The connections to physics begin to emerge in this chapter with a brief but very pleasant discussion of the Euler-Lagrange equations and the classical variational problems of mechanics.

Chapter 3 breaks suddenly with classical differential geometry with a presentation of the elements of point set topology and smooth manifolds. The smooth structure of differentiable manifolds, discussed with well-chosen pictures and a minimum of abstraction.. Orientation — a topic that often gets bogged down in technicalities — is presented clearly, defined in terms of local coordinates of overlapping charts on a manifold M. From here the emphasis on basic linear algebra becomes very pronounced indeed and anyone hoping to get by with a mere smattering of matrix manipulating skills is bound to get lost. Which is fine by me.

Chapter 4 covers the essentials of Riemannian and semi-Riemannian manifolds. The central concept stressed here is the metric tensor. Affine connections and covariant derivatives are discussed, mostly in terms of the Christoffel symbols. Vector bundles are described along with symmetric connections on M. The general curvature tensor is then described. The author wisely gives visual examples, such as parallel translations along the side of a square in Euclidean space, in this section. The section closes with geodesics on M and their equivalence to solutions of the general Euler-Lagrange extremal equations on M.

Chapter 5 and 6 deal with matters rarely seen in general differential geometry texts: an elementary but fairly complete introduction to the Lobachevskian plane H and Minkowski spaces as semi-Riemannian manifolds in chapter 5 and a very geometric discussion of minimal surfaces in chapter 6. Both of these chapters make heavy use of elementary complex analysis. Front and center are the group PSL(2,R) of Möbius transformations. This leads to the classic characterization of the geometry of H: Möbius transformations are orientation-preserving isometries. The role of this space and its generalization, Minkowski space, in special and general relativity is then discussed. Chapter 6 continues the use of complex analysis with the examination of conformal parametrizations of surfaces and the proof that all two-dimensional Riemannian manifolds are conformally Euclidean regardless of metric. This is one of the most visual chapters in the book.

The rest of the book is a collection of more advanced topics. Chapter 7 gives a presentation of the basics of Lie groups and algebras and their roles in differential geometry and physics, including Pauli matrices and quaternions. Chapter 8 is a refreshingly elementary account of the representation theory of Lie groups and algebras. In these chapters, the author is laying the foundation for a study of quantum and classical mechanics and electromagnetism on Possion and symplectic manifolds through Hamiltonian systems in the next and final chapter. Any serious physics major struggling with these concepts would benefit immensely from studying these chapters.

This is not a book for the dabbler. A lot of material is covered, and it is covered seriously. The interplay of physics and geometry comes into play a great deal. The material is quite visually presented; in fact, there is a surprising number of pictures for a book at this level, always a plus in a geometry book.

Most of the exercises are straightforward. What I did find a bit problematic about the exercises was that most of them are given in the first six chapters. This is probably due to the “optional” nature of the last three chapters, but still — what good is having all this wonderful material without exercises to test and develop the student’s understanding? There are overall fewer exercises than one would like. In a book of this level, supplementary problems at the end of each chapter would have helped.

At times Taimanov gives a rushed presentation, drawing pictures rather than proving a result or stating a definition carefully. This is particularly noticeable in chapters 1 and 2, where the resulting presentation of classical differential geometry isn’t nearly as careful or complete as it should be. The presentation of many of the same concepts in MGSF is much more careful and detailed, and hence a lot easier to follow.

A much bigger problem — especially for mathematics majors — is the very heavy emphasis on tensor algebra. Consider the discussion of the Gauss-Bonnet theorem in chapter 2. A lot of professors avoid the notational nightmare of the Christoffel symbols. This is understandable but not very helpful for this particular result, which is so critical to surface theory. Unfortuately, this is the only place in the book where the emphasis on tensor methods is a help and not a hindrance. I found his initial discussion in chapter 3 of basic tensor algebra to be a bit murky even with the initial example in the previous chapter. That’s when I knew it was going to be a problem. From that point on, Taimanov uses tensor notation much more heavily and it begins to bog down the presentation, particularly because he uses the Einstein-Ricci indices notation. I understand why physicists rely so heavily on this notation in their computations and I can also understand the author’s choice here as he wants to keep the presentation as accessible as possible for the physics students. But there’s a reason why nowadays most mathematicians avoid using it when defining these concepts and explain it only when the algebraic framework is constructed first. For example, on page 58 Taimanov defines a covector as

an object given in local coordinates {x_aⁱ} by an ordered collection of numbers (v₁^a, … ,v_aⁿ) and the corresponding collection (w₁^b, … ,w_bⁿ) for another coordinate system {x_bⁱ} satisfies the equation: w_j^b = (∂x_aⁱ/∂x_bⁱ) v_i^a

Uh, ok. I think most mathematics students reading this are going to have their eyes glaze over. The definition of a covector as an element of the dual space of the tangent space of M is more sophisticated, but I think in the long run it provides greater insight into this machinery and opens the door for deeper understanding. This is a case where a well-intentioned attempt at keeping things simple actually muddies the waters further. Modern precision may require more effort, but it pays much bigger dividends in overall comprehension.

The big question is who is the target audience of this text. To be able to read the first 5 chapters, you need to be pretty comfortable with calculus (construed generally enough to mean a careful presentation a la Spivak’s Calculus) and you better be very comfortable with elementary geometry using Euclidean transformations and linear algebra up to and including Hermitian matrices and bilinear forms. After this, a good grounding in group theory is needed as well as some experience with complex analysis and differential equations (particularly in the last two chapters). In Europe or Russia, this book could probably serve upper level undergraduates well — but not in the US.

That being said, however, this is a wonderfully diverse and challenging introduction to one of the central topics in both mathematics and physics. It’s certainly a terrific text to keep handy as a supplement when teaching the subject. And of course, it would a very handy text for graduate students come prelim time.

Andrew Locascio is a second year graduate student at Queens College of the City University of New York. His interests are differential geometry, topology and the relationship of mathematics to the physical sciences.