I closed my 2013 review of Part I of this duo with the phrase, “This very good book makes one impatient for the appearance of the second volume!” and now here it is. In the Introduction to Part I, Rudolph and Schmidt provided some foreshadowing of Part II with the promise that “The second volume will deal with fibre bundles, topology and gauge field theory, including aspects of the theory of gravity.” Thus, if it is at least reasonably fair to say that Part I is focused on differential geometry in a somewhat more classical sense, with Lie theory and Hamiltonian dynamics on display center stage, Part II quickly gets more Riemannian — and then some.

While differential geometry is still the order of the day, now the perspective is rather different than before, and the main player at the start of this game is Hermann Weyl, the man responsible for the deep idea of gauge invariance. The authors say this about Weyl’s work at the beginning of the present Introduction:

The concept of gauge invariance first appeared in … papers of Hermann Weyl from the year 1918. In this work, Weyl extended Einstein’s principle of general relativity by postulating that, additionally, the scale of length can vary smoothly from point to point in spacetime. In more detail, Weyl’s basic idea was to develop a purely infinitesimal geometry. Behind that concept was his belief that ‘a true infinitesimal geometry should … recognize only a principle for transferring the magnitude of a vector to an infinitesimally close point …’ [And so] … the notion of connection appeared for the first time in the mathematical literature.

This is a wonderful way indeed to introduce what lies ahead in the roughly 800 pages of Part II, supplementing Part I’s beefy 700 pages: yes, there is the old joke about the German default literary setting of encyclopedic coverage, but it’s entirely right to spend this much space and time on the given material — and, yes, Part II is also fabulous.

Before I get on to other things, let me note here how on target the authors’ approach is for mathematicians as opposed to physicists: the notion of a gauge, i.e. of a gauge theory, located at the heart of so much of modern physics, is perhaps off-putting to mathematicians should they first encounter it in the way it is presented by physicists: that was certainly my experience, and I gather that mine was not an isolated singular case. But there’s a royal road to gauge theory: none other than Sir Michael Atiyah says at the start of the first chapter of his (gorgeous) book, *The Geometry and Physics of Knots*, that

[a]fter a long fallow period in which mathematicians and physicists pursued apparently independent paths, their interests have now converged in a striking manner. However, it appeared that parallel problems were being investigated in the past but a common language and framework were missing. This has now been rectified with gauge theory (alias the theory of connections) providing the common ground.

The point is that for us mathematicians this notion, central to modern physics and geometry as it interfaces therewith, really should be fitted into the framework of Riemannian geometry, i.e. the theory of connections, and it’s an additional gain if the presentation is historically sound, showing its mathematical evolution over time. Even if the physicists present such themes more pragmatically, oriented to getting at the physics proper as soon as possible, such an approach is obviously inimical to the mathematical ethos, and it is comforting indeed to note that Hermann Weyl, despite his importance to physics, was first and foremost a mathematician. Gauge theory’s mathematical pedigree is sound.

It is well-known that the model *par excellence *of a gauge theory in physics is the theory of electromagnetism, with Maxwell’s equations running the show. It is also the case that the standard model is rife with gauge theory: the theories of three of the four fundamental forces (add the weak and strong forces to the list) are dominated and directed by gauge groups. It is part and parcel of grand unification that at high energies these gauge interactions merge into a single force (yes, this is a Wikipedia characterization), and the challenge is to fit gravitation into the scheme.

This is consonant with the layout of the book under review in that (in contrast to their estimate in Part I, cited above) gravitation is out, but everything else is in. By the way, this refusal on the part of gravitation to fall in line with other gauge theories is historically fascinating, given that, as Rudolph and Schmidt indicate in the above quote, Weyl’s introduction of gauging goes back to his attempt to go beyond Einstein’s general relativity. In point of fact, they go on to give the relevant details themselves:

[Weyl] obtained a unification of general relativity with electromagnetism. However, it quickly became clear that the model was not compatible with basic physical principles. It was Einstein who observed that if this theory was correct then the behavior of clocks would depend on their history. This is in contradiction to empirical evidence.

But[!] “in 1929 Weyl proposed to apply [the gauge principle] to quantum mechanics …” and the rest is history, both not so recent and recent: if the 1-dimensional unitary group is picked as the gauge group we get nothing less than quantum electrodynamics (from Dirac in the 1920s to Schwinger, Tomonaga, and Feynman (and Freeman Dyson!) in the late 1940s). Then in 1954, Yang and Mills proposed the first non-abelian gauge theory (with *SU*(2) coming into play). The authors conclude these very evocative introductory historical remarks with allusions to work dating to the 1960s and 1970s, including the famous achievements of Weinberg and Salam on electroweak interactions and Gross-Wilczek, Politzer, and Weinberg on quantum chromodynamics.

Well, so much for the physics. What about the mathematics? Rudolph and Schmidt draw a trajectory from Weyl and Élie Cartan (with the theory of differential forms being another natural starting point for the whole business, in contrast to what Weyl was up to, starting with the absolute differential calculus of Ricci and Levi-Civita, already familiar of course from Einstein’s own approach to general relativity) to Ehresmann, Seifert, Whitney, Hopf, Stiefel, Whitney, Hurewicz, Steenrod, Pontryagin, and Chern; in other words, to the theory of fibre bundles and characteristic classes. And there it is: we have the flowchart, so to speak, for the book. To wit: the first four chapters deal with fibre bundles as such and go all the way to cohomology and characteristic classes; after this it’s Clifford algebras, Yang-Mills, matter fields, and pretty sophisticated stuff on the notion of a gauge orbit space and quantum gauge theory (closing the proceedings).

So Rudolph and Schmidt certainly provide some very powerful physics as the culmination of their impressive geometric efforts, but the geometry itself is by itself well worth the price of admission. I first encountered these notions in masterful presentations by Milnor (and Stasheff): there is a fabulous crash-course in Riemannian geometry available in Milnor’s unsurpassed text, *Morse Theory* (see also the book’s review by Nicolaas Kuiper) and characteristic classes are dealt with definitively in the book by that title, written by Milnor and Stasheff (reviewed by Edwin Spanier). It is truly very nice to have these themes dealt with in the present uniform context by Rudolph and Schmidt, with a particular focus of gauge theory orchestrating everything.

It really goes without saying at this point that Part II of *Differential Geometry and Mathematical Physics* is a very important pedagogical contribution and a worthy complement to Part I. It presents fine scholarship at a high level, presented clearly and thoroughly, and teaches the reader a great deal of hugely important differential geometry as it informs physics (and that covers a titanic proportion of both fields). Additionally, Gerd Rudolph and Matthias Schmidt do a fabulous job presenting physics is a manner that mathematicians will not find *unheimlich*. *Ausgezeichnet*.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.