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Selected Papers I, Shiing-Shen Chern

Shiing-Shen Chern
Publication Date: 
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Springer Collected Works in Mathematics
[Reviewed by
Michael Berg
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For many years I've been the owner of a hard-cover copy of volume I of the set of books under review, and now I have it double, plus the remaining three volumes — and this is still only a selection of Chern's output, which testifies both to Chern’s fecundity and to his influence on mathematics, specifically differential geometry. It goes without saying therefore that this collection of 147 – 29 = 118 papers (147 being his total output, with 29 omitted in this collection), spread out over four volumes, is a proper centerpiece for the library of any geometer. In fact, even for others like me, who have occasional need for differential geometry and are enthralled with its beauty and depth (and pervasiveness!), the set is very attractive indeed, given that Chern is such a wonderful and clear expositor. As an illustration, two years ago I had occasion to lecture on, and later use in a publication, material on sheaf cohomology as developed by Serre and (most emphatically) Grothendieck, including a proof of de Rham’s theorem, and I needed a source that was much more user-friendly and accessible than Grothendieck’s original development. I hit on a truly marvellous and elegant discussion of all the relevant material by Chern in his 1967 book, Differential Geometry without Potential Theory, and was especially struck by the way Chern trimmed away all but the non-negotiable necessities and presented a very smooth and clear discussion, all in the context of complex manifolds. This high-level mastery of material and style is rare, of course (one thinks of Serre or Bott), and it is irresistible. It is no wonder, therefore, that his reputation as a very great geometer was complemented by his reputation as a superb and inspirational teacher.

There’s a short on-line presentation, namely, , dealing with Chern’s life and his impact in several dimensions, to coin a phrase, and one is immediately struck by, for example, the historical angle, with Chern being a disciple of Blaschke as well as of Elie Cartan (from whom Chern learned, for instance, web geometry and the method of moving frames, respectively). From a purely mathematical angle, it is awe-inspiring that Chern’s contributions include the Gauss-Bonnet theorem (in its modern form for differentiable manifolds), and of course Chern classes: centerpieces par excellence of modern geometry.

S. S. Chern was indeed one of the central figures of twentieth century mathematics and is counted as nothing less than the main architect of modern differential geometry in the wake of Elie Cartan. This collection of 118 of his papers is an appropriate tribute to him and his work. Finally, I particularly recommend the opening essay, “S. S. Chern as geometer and friend,” by André Weil as both a personal reminiscence and a fine appraisal of Chern’s work. It is a very good way to introduce this wonderful collection. 

See also volumes two, three, and four.

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

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