I’ve been a graduate student over a year and half now and it never ceases to amaze me what a different world it is from being an undergraduate. There are still classes, there are still grades, there are still professors you like and professors that make you want to hire a hit man. But there’s a deeper sense in which it’s a completely different world. The main thing is the far greater independence that’s expected of you. You’re required to learn a lot on your own. There are of course caring professors and fellow graduate students who will help. But if you try and take a course in algebraic geometry and you don’t know basic category or module theory, you’re pretty much dead meat.

It’s hard to see how the system can be otherwise. Graduate students are attempting to make the transition to being *professional mathematicians*. As such, they will need to learn new mathematics for their own work quickly and with no guidance. The intent at many programs is to race graduate students to the research level as quickly as possible at all costs. Thus, a lot of graduate programs encourage students to learn only what they need for their work. So graduate students need books that are concise but don’t lack any necessary details, books that cover their subjects in a broad and up-to-the-minute way. Ideally, the work should also touch on connections with other areas of mathematics.

There is a general dearth of such works — what one mostly finds are bags of definitions with the entire course laid out for the student to build. At the other end of the bell curve are gigantic reference works like Serge Lang’s *Algebra,* John Lee’s *Introduction To Smooth Manifolds *or David Eisenbud’s *Commutative Algebra With A View Towards Algebraic Geometry* — books which are simply too large to be effectively used except *very *selectively.

Which brings me to Jürgen Jost, a mathematician and author who seems very sympathetic to the graduate students’ plight and has set about writing texts appropriate for such circumstances. Jost is one of the world’s eminent applied analysts. He been teaching courses to graduate students at the Max Planck Institute in Leipzig for nearly three decades and has a reputation as a first rate teacher. This reputation is well-supported by his half-dozen textbooks and research monographs for advanced students and researchers. The sheer range of topics in modern analysis, geometry and mathematical physics they cover is astounding: *Postmodern Analysis*, *Compact Riemann Surfaces*, *Calculus of Variations*, *Nonpositive curvature: Geometric and Analytic Aspects, Partial Differential Equations *and several others. He’s even begun work on a graduate text on mathematical biology; a preliminary version can be found at Jost’s web site.

All of Jost’s texts are very much in the old “Göttingen”style — comprehensive, concise without sacrificing any important definitions or theorems, self-contained, and giving the state of the art in each topic. They are quite abstract, but all contain many motivating examples. The goal of these texts is to provide a solid basis for advanced students to begin mathematical research upon mastering the material contained therein. The text under consideration here — *Riemannian Geometry And Geometric Analysis*, 5^{th} edition — is completely in this spirit and a very worthy addition indeed to Jost’s textbook oeuvre.

With the possible exception of complex analysis, differential topology and geometry may be the subjects for which there is the greatest choice of quality graduate texts. It is, after all, one of mathematics’ most inviting subjects: it requires minimal prerequisites, it is inherently visual due to the Whitney Theorem and, best of all, it is one of the most applicable of mathematical disciplines. Indeed, one can make a very good case that most mathematical models in both classical and modern physics ultimately rest on the analysis of a differentiable manifold.

Most texts written since the 1960s they tend to emphasize the topological and geometric aspects of the subject by using coordinate-free methods, downplaying local coordinates and the fearsome “debauch of indices” common in earlier texts. The study of differential manifolds has, however, moved in a considerably different direction since the writing of Spivak’s and Milnor’s classics. Analysis has come to the fore. Hamilton’s Ricci flow and variational analysis on semi-Riemannian manifolds have solved deep problems in both geometry and physics, culminating in Perleman’s proof of Thurston’s Geometrization Conjecture and consequently of the Poincaré Conjecture.

Many of these results have been the product of the rapid expansion of *geometric analysis*, which can be roughly described as the study of partial differential equations and systems of ordinary differential equations on manifolds and submanifolds of the same dimension as the solution spaces. The solution of such equations, particularly in equations of elliptic type such as the Euler-Tricomi equation, yields geometric information on the structure of the manifold. Conversely, aspects of the topology of the manifold yield analytic data on the PDEs. Since geometric analysis by its very nature requires differential equations expressed in local coordinates and the analytic tools needed to solve them, coordinate free concepts are of very limited use. Therefore, most of the standard introductory texts have avoided the subject altogether. The upshot of all this is that graduate students that study differential manifolds in the conventional manner will be very ill equipped to study the current literature.

Jost’s book attempts to remedy this by presenting an integrated treatment of both basic Riemannian geometry and geometric analysis in a single course. This not only makes the presentation more contemporary than most texts, but also covers more topics closer to the frontiers of research. The book is heavily computational; just about everything is expressed primarily in local coordinate tensor notation. This is differential geometry seen through the perspective of an analyst. So, in a sense, Jost has written a very modern book in an old fashioned manner. One would think this would result in a text that is nearly unreadable. His great triumph here is he succeeds in crafting a book that is anything but. He uses an absolute minimum of expository prose to produce the absolute maximal clarity. This results in *one hell of a book,* especially for serious graduate students.

Chapter 1 gives — in 89 pages — a remarkable short course in differential manifolds and Lie algebras. Jost’s own words best describe what he has accomplished in the first chapter:

In the first chapter, we introduce the basic geometric concepts, like differentiable manifolds, tangent spaces, vector bundles, vector fields and one-parameter groups of diffeomorphisms, Lie algebras and groups and in particular Riemannian metrics. We also treat the existence of geodesics with two different methods, both of which are quite important in geometric analysis in general. Thus, the reader has the opportunity to understand the basic ideas of those methods in an elementary context before moving on to more difficult versions in subsequent chapters.

In this first chapter, Jost frequently uses concepts from algebra to build smooth structures on manifolds. This is a very natural thing to do, but it requires instantaneous grasp of the notions of linear spaces and their isomorphisms. (In German universities, undergraduates are expected to develop expertise in abstract algebra early. This is a lesson we can certainly take to heart for our students.) He uses the concept of local coordinates to explain the Einstein summation convention. It is the clearest statement I’ve ever seen. Physics majors struggling to understand this convention can do a lot worse then read this chapter. The inner product property of the Riemannian metric is emphasized; this is in keeping with the analytic flavor of the text and makes a lot of the concepts later easier to digest. The chapter concludes with a wonderful introduction to the Lie group Spin(V).

The second chapter introduces de Rham cohomology and related essential tools from elliptic PDEs. He proves the existence of harmonic forms, a point not often emphasized. The presentation of each concept leads to later ones as the discourse deepens. Restricting the elements of the cohomology classes to linear maps only allows the reader to understand some critical ideas of their general study, such as the Hodge lemma and the de Rham cohomology group on Sobolev spaces, without the technical difficulties of nonlinear problems a fully general presentation entails. Despite the fact that Jost is careful to define the de Rham cohomology and its relevant machinery in a completely self-contained manner, I get the distinct impression he expects the student to have at least a passing acquaintance with cohomology. He provides a separate appendix on basic topology — underscoring how adamant Jost is about making the book self contained.

The third chapter treats the general theory of linear connections and curvature. About half of this chapter is standard: parallel transport, covariant derivatives, the Bianchi identity, the Levi-Civita connection and flat connections. The remaining material is extremely nonstandard and will be of immense benefit to both analysis and physics students: an introductory discussion of the Yang-Mills operator using characteristic classes, as well as discussions of the Bochner method for harmonic forms of nonnegative Ricci curvature, connections on Spin(V), and the Dirac operator.

In the fourth chapter, he gives a fairly complete development of the theory of Jacobi fields and the related machinery. This includes conjugate points, distance minimizing geodesics, and, most importantly, a discussion of Riemannian manifolds of constant curvature. This sets the stage for proving the Rauch comparison theorems for Jacobi fields, applying these to deriving the usual Jacobi estimate inequalities. The chapter concludes with a development of the global geometry of spaces of nonpositive curvature and a proof of Reshetnyak’s quadrilateral comparison theorem.

In between Chapters 4 and 5 is a very significant “intermission” which states many results of global differential geometry used in the last three chapters and which form the basis of many of the “standard” texts on Riemannian geometry like do Carmo or Petersen. Many of these results are stated without proof. Among these are the global Gauss-Bonnet theorem and the convergence theorem. However, there is also a brief introduction to the current literature on Ricci flow and related analytic results such as the Finiteness Theorem. Therefore, the entire current structure of Riemannian geometry is summarized in this chapter, with complete references. There is probably no such summary anywhere else in the textbook literature. Frankly, if this was the only chapter the book contained, it would still be worth having.

With these results in hand, Chapter 5 gives a very compete and up to date treatment of the complex projective space; constructing Kahler manifolds and symmetric spaces as important examples of Riemannian manifolds Their geometry and associated Lie algebras are constructed in full detail, first in the compact case and then in the noncompact case. The noncompact case is developed in full as an example of a class of nonpositively curved Riemannian manifolds. I know of no other general textbook that does this except for Daniel Huybrucht’s *Complex Geometry: An Introduction* (2004).

Chapter 6 is about one of geometers’ favorite subjects: Morse theory. The usual topics are discussed with more than usual clarity: saddle points on manifolds, nondegeneracy of critical points, the Morse Lemma, stable and unstable manifolds, foliations and more. There are also more modern topics discussed such as graph flows and general orientations for manifolds at local extrema. This chapter also breaks strikingly with the rest of the book: it’s the only chapter with many pictures. That’s only natural — Morse theory and the Floer homology are such beautifully visual topics that a purely formal treatment would lose a lot of their full significance. It is also the first place in the book where infinite dimensional manifolds are briefly mentioned and defined. The main differences between them and the finite dimensional case are discussed in preparation for the Palais-Smale compactness condition and proving the existence of closed geodesics. This sticks to Jost’s overall philosophy:“If we don’t need it, we don’t mention it.”

Chapter 7 discusses the behavior of harmonic maps on general Riemannian manifolds by means of the energy density of local coordinates at a point on such manifolds. This is a mainstream topic of geometric analysis; one of its main tools the Hamilton flow. This includes the higher regularity estimates and the generalized Bochner technique as well as the behavior of such maps on manifolds of nonpositive sectional curvature. Chapter 8 extends these results to multivariable harmonic maps defined on Riemann surfaces. The book closes in a blaze of glory with a chapter on variational problems in quantum field theory. The machinery developed in the last four chapters is brought to bear on three of the most important functionals in string theory: The Ginzburg-Landau functional, the Seiberg-Witten functional and the class of Dirac-harmonic maps. These are developed as purely mathematical objects, but Jost is careful to point out differences in notation and the close relationship to physical models such as the nonlinear super-symmetric sigma model.

The amount of material Jost succeeds in covering in 589 pages is positively astonishing, and he manages to do it completely coherently. Obviously this results in a book which is incredibly dense. The book is broken into bite-sized pieces, which helps a lot. Even so, the pace of the book can only be described as relentless. What’s incredible is that despite that and the level of difficulty, Jost never ceases to hold the reader’s interest and remains lively and fascinating.

Each chapter builds on the ones before it, so that the chapters are never separate entities. One of the ways Jost does this is with wonderful sections at the end of every chapter subsection, called “Perspectives”. These broaden the vision of the text even further with brief digressions into the history and current state of the central topic as well as exhaustive references for further study. As a result, the book actually ends up covering a lot more then it seems to, making it that much *more* impressive.

One part of the book that will appeal most to readers is that Jost gives *lots *of examples for every concept. Early in the book, these are given separately in batches of a dozen or so. As the concepts become more sophisticated, as in the variational analysis chapters, theorems are proven first in a special case and then if possible, generalized. This really helps with difficult concepts. Otherwise, the book’s merciless pacing would just obscure everything.

I have just two minor complaints and a concern. First my usual complaint with books on geometry at this level: there aren’t enough pictures. The only place where pictures are plentiful is in the chapter on Morse theory. One of my friends says that it’s better for graduate students to draw their *own *pictures. He is probably right; even so, Jost should have put a few more pictures in the book, particularly in the early chapters when most of the examples are easily visualizable. My other complaint is that there are not enough exercises. This seems to be a general problem with European textbooks.This may be inherited from the old days when the exercise sets to European texts usually came in a separate booklet, as in Courant’s original calculus lectures.

This is where my concern comes in. Despite my wholehearted endorsement of the text and the great effort Jost has put into making it self-contained, the book’s sheer density and level of difficulty would make me hesitate to use it all by itself. *Can* the book be used by itself for a graduate differential geometry course? Certainly. With a good teacher, any graduate student can use it to learn a *lot *of modern geometry *very* quickly. The problem is that the book has a very clear agenda and it has to focus totally on this agenda. As a result, a lot of very important topics are omitted or are stated without proof in the “intermission.” This really makes the book nearly impossible to use for self study by itself for any but the very best students at the very best schools. But since Jost is so diligent about making his students familiar with the literature, choosing supplementary material will not be difficult at all.

These are really minor issues. Jost has produced another jewel in a crown of wonderful graduate level texts. An outstanding year-long course in differential geometry at the graduate level can be built on it. Here’s how I would do it: The first semester could begin with Jost’s first two chapters combined with Loring Tu’s *An Introduction to Manifolds, *which would give students a complete and very up-to-date course in differentiable manifolds. The second semester would then cover chapters three to eight in combination with some of the recommended papers in the “Perspectives” sections and Peter Petersen’s *Riemannian Geometry*. This would make a very strong course in Riemannian geometry and geometric analysis. All these books are relatively short and between them, they cover just about everything any graduate student could want or expect to learn about modern differential geometry. I certainly plan to spend a great deal of time going through Jost’s book slowly in the very near future.

Andrew Locascio is a second year Master’s student at Queens College of the City University of New York. His interests are topology, noncommutative algebra, additive number theory and the relationship between mathematics and the physical sciences. He plans to begin applying to PhD programs in pure mathematics this November. The answers to any other questions regarding him can be found at his Facebook page at http://www.facebook.com/home.php#/home.php?ref=home. He always wishes to hear from his fellow mathematicains or aspiring ones, if only so he doesn’t feel so alone in a world of practical creatures.