Differential Geometry & Integral Geometry

It is surely one of the great understatements of the age to say that differential geometry is hot stuff. Only around a decade ago the Poincaré conjecture was settled in toto by Perelman, with Hamilton’s Ricci flow taking center stage in the proof. Furthermore, there is an all but unprecedented cross-fertilization still going strong between differential geometry and physics, specifically centered on such things as quantum field theory and string theory, even to the point of encyclopedic tomes like Quantum Fields and Strings: A Course for Mathematicians hitting the open market. On the quantum field theory end, I personally find the whole business surrounding Segal and Atiyah’s axiomatic quantum field theory entirely irresistible, given that due to Witten’s work of the late 1980s, it is provides one with a way of getting at the Jones polynomial (yes, from knot theory and therefore low-dimensional topology) by means of a yoga of Feynman integrals. Surely, this is about as ecumenical as mathematics gets. Agreed, once Feynman integrals enter the game, we mathematicians find ourselves properly ridden with anxiety and misgivings, but this is still exceedingly exciting material, rigor questions notwithstanding. (A propos, here is a fascinating Briefwechsel dealing with these issues: http://www.ams.org/journals/bull/1993-29-01/S0273-0979-1993-00413-0/S0273-0979-1993-00413-0.pdf, followed by http://www.ams.org/journals/bull/1994-30-02/S0273-0979-1994-00503-8/ )

Let’s very briefly take up the theme of topological quantum field theory and knot theory — and, by the way, do read Atiyah’s The Geometry and Physics of Knots for the sheer beauty and elegance of it all. The context in which all this takes place is Chern-Simons theory, which is a Yang-Mills (gauge) theory, and we all know how sexy gauge theories are to any modern physicist. Note, however, that gauge theory originated with a mathematician, Hermann Weyl, when he decided to do some general relativity, and, as Atiyah remarks in loc.cit., when physicists speak gauge, they’re actually speaking of connections in principal fibre bundles, and the Lie groups for what we’re dealing with are generally of the SU(n) flavor. So, indeed, this is ultimately differential geometry with a vengeance.

And so we have dropped Chern’s name already: the paper in which all this great stuff first appeared was co-written by Shiing-Shen Chern and Jim Simons, now also known to the real world as a hedge-fund billionaire. All this paints an evocative picture: after all, Simons worked with Chern at UC Berkeley in the 1960s, where Chern was building and leading the world’s premiere school of differential geometry, his PhD students including, for instance, Manfredo do Carmo in 1963 and Shing-Tung Yau in 1971. The book under review presents the reader with nothing less than Chern’s own notes for his foundational advanced courses, as well as a number of very important and useful expository articles.

Specifically, the present book is an amalgamation of two sources, and here is what the editors say about the first:

the first book, Topics in Differential Geometry, comes from the title of his lecture notes at IAS … in 1951 [and] also contains his expository papers … “From Triangles to Manifolds,” “Curves and Surfaces in Euclidean Space,” “Characteristic Classes and Characteristic Forms,” “Geometry and Physics,” and “The Geometry of G-Structures,” together the set of so far unpublished lecture notes: “Minimal Submanifolds in Riemannian Geometry.”

This reads like what it is: a trailblazer’s careful account of his work and discoveries soon after the dust has settled; it is also a pedagogical gem in that we have here before us the material Chern himself crafted for the purpose of teaching his geometry to his apprentices. Additionally, the inclusion of heretofore unpublished material is obviously a fabulous event and the editors are right to announce it in italics. They add, very much on target: “… [the contents of the first book] show how differential geometry is connected to other subjects such as topology and Lie group theory [and although] there are more modern expositions of these topics, they are usually not comparable with what Chern wrote.” They conclude that “[the] first book will be very valuable to beginners [in] modern differential geometry, and will also be very valuable to experts.” It is impossible to disagree with this appraisal.

But it gets even better: “The title of the second book is a combination of titles of two sets of lecture notes … which have not yet been formally published. It seems that there exists only one known copy of the second set … owned by the library of the University of Michigan. It also contains a more recent unpublished set of lecture notes titled ‘Lectures on Differential Geometry.’” Again, this material is obviously of both historical and pedagogical interest, not least because of the fact that Chern’s track record as an incomparable teacher matches his towering reputation for groundbreaking research in the field, indeed, a field he essentially defined.

There is not much that needs to be added: this is a fantastic addition to the mathematical literature.

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