Shen Chern Selected Papers Volume 4

Working Through Chern's Later Papers

I ran into an issue last year while trying to track down a specific argument from Shen Chern Selected Papers Volume 4 that referenced a lemma in his 1974 paper on gauge connections. The citation in the book was slightly off from the original journal version, which caused me to waste about three hours cross-checking equations before I realized the formula in the collected volume had a sign typo that the original didn't have. The workaround was just going straight to the Proceedings of the National Academy of Sciences archive for the primary source. Once I confirmed the correct form, everything aligned properly. This volume mainly covers work from the mid-1970s through the early 1980s, which is where Chern shifted focus toward problems in mathematical physics and the geometry of fiber bundles. If you're approaching these papers as a graduate student, the jump in style from his earlier classical differential geometry work is noticeable. The notation gets denser, and he assumes familiarity with background material that he never bothers to review. That's not a flaw in the writing; it's just how his mind worked at that stage. He stopped explaining the preliminaries because he no longer needed to.

Why You Should Use the Original Rather Than a PDF Scam

There are a lot of sketchy sites offering cracked PDFs of this volume, and I'd strongly advise against them. The scanned pages often have corrupted footnotes, missing indices, and misaligned equations. For a text this mathematically precise, a single bad character in a formula can send you down a completely wrong path for days. The publisher, World Scientific, sells the hardcover directly, and university libraries usually carry it. The cost is steep if you're buying standalone, but most institutions have it in their math stacks already. The actual papers inside cover topics like characteristic forms, the Chern-Simons functional, and several collaborations with S.-S. Chern's collaborators on curvature and topology. One thing beginners consistently miss is that these papers are not meant to be read cover to cover like a textbook. Chern was writing research notes, often with other authors, and the results are scattered across different technical levels within the same volume. You pick the thread you need and trace it backward through the references. The index at the back helps, but it's not comprehensive enough to rely on. A counter-intuitive point: the papers here are actually more accessible than his earlier collected volumes if you already know the basic language of connections and curvature. In his later work, Chern stopped working in local coordinates as much and leaned heavily on invariant formulations. That means fewer pages of tedious computation and more emphasis on the structural relationships. If your background is weak in fiber bundle theory, go back and solidify that first. Attempting these papers without a working knowledge of principal bundles and characteristic classes will just frustrate you.

The main downside of this volume is that several of the papers assume you've already read the corresponding earlier papers from Volume 2 or Volume 3. Chern referenced his own results frequently without restating them. If you're jumping in cold, you'll hit gaps quickly. There's also the problem of notation drift between volumes. He wasn't consistent about redefining terms across papers, which means a symbol that meant one thing in 1975 might mean something subtly different in 1981. For a concrete example of how this plays out, in the paper on secondary characteristic classes, Chern reuses the symbol theta to denote both a connection form and a curvature-related object depending on the section. If you don't track which usage applies in each paragraph, the calculations become opaque. I keep a personal notation reference sheet when working through these, mapping each symbol to its definition as I encounter it. It takes maybe twenty minutes upfront per paper and saves hours of confusion later. If you need the material but can't access the physical volume, the closest legal alternative is checking whether your institution has the World Scientific set through a digital subscription. Some universities have licensed it through platforms like EBSCO or JSTOR, though coverage varies. Another route is the Mathematical Sciences Research Institute, which sometimes hosts archival materials related to Chern's work, particularly around the programs he directed in the 1990s.

Bottom line, this volume is essential reference material for anyone doing serious work in geometric analysis or topological field theory, but it demands that you come prepared with the right foundations and the patience to navigate its idiosyncrasies. It won't hold your hand, and it won't substitute for reading the primary sources carefully. That said, the mathematical insight packed into these pages is genuinely unmatched.