In graduate school, my ruefully inadequate exposure to differential geometry entailed a half-hearted attempt at learning a reasonable amount of basic material from Boothby’s Introduction to Differential Geometry, still a standard text at that level. In the full flower of youthful arrogance and goofiness I figured that I should never have any need to know this material in any real depth: why should an algebraic number theorist learn any differential geometry?
I was wrong, of course, as I have come to realize all too well in later life. My work in number theory has developed in directions requiring a real measure of facility in such areas as homological algebra, sheaf theory (and sheaf cohomology), and, yes, most recently, differential geometry. While it is part and parcel of mathematical scholarship to devote a good deal of time and energy to study, at any time of one’s career, it is unavoidable to chastise oneself for such sins of youth. But crying over spilt milk won’t do, so, after several decades of studying and working in number theory, I now find myself faced with the task of having to learn a substantial amount of material about such things as fibre bundles, vector bundles, connections, Riemannian manifolds, and differentiable structures. Walter Poor’s Differential Geometric Structures, first published in 1981, now re-issued by Dover, is a God-send for this purpose.
Indeed, the book under review looks to be perfect for self-study, and, possibly, for a second course in differential geometry. Poor points out that some background on the part of the reader is assumed, and he cites the aforementioned text by Boothby as a good prerequisite. But Differential Geometric Structures is so well-written that a sufficiently mathematically mature reader, even without formal differential geometric preparation, should be able to make his way through the book without too much difficulty: the prose is compact and clear, there are good examples, and there are good exercises. Additionally, Poor provides superb brief motivations at the start of most of his chapters, bringing the upcoming material into focus from the outset, and making for a wonderful flow of the narrative that provides the reader with a good (early) sense of what is going on and what it is all supposed to be about.
As regards the contents proper of the book, Poor starts off with fibre and vector bundles and, if you’ll pardon the pun, Riemannian connections, and then goes on to Lie theoretic aspects, symmetric spaces (which rings a number theoretic bell: there is tie here with automorphic functions), and some more specialized material: symplectic and Hermitian vector bundles, more on (“other”) geometric structures. Good stuff!
I think this is a wonderful book and look forward to studying it myself in the near future. It is a pleasure to recommend it in very strong terms.
Michael Berg is Professor of Mathematics at Loyola Marymount University in California.