by Hung-Hsi Wu
Shiing-shen Chern, one of the giants in the history of geometry, passed away in Tianjin, China, on December 3, 2004, at the age of ninety-three. He had spent the last five years of his life in Tianjin, but his career was established mainly in the U.S. from 1949 to 1999.
Chern’s work revitalized and reshaped differential geometry and transcendental algebraic geometry. In the decades before 1944 when he embarked on the writing of his historic papers on Chern classes and the geometry of fiber bundles, the field of differential geometry had gone through a period of stagnation. His papers marked the re-entry of differential geometry into the mathematical mainstream, and his tenure at Berkeley (1960–1979) helped make the latter a premier center of geometry in the world. In his lifetime, he had the pleasure of seeing Chern classes become part of the basic vocabulary in contemporary mathematics and theoretical physics. There are also Chern–Weil homomorphism, refined Chern classes for Hermitian bundles, Chern–Moser invariants, and Chern–Simons invariants. Chern was also a mathematical statesman. One does not often see great mathematical insight and great political leadership converge on the same person, but Chern was that rare exception. He was the main liaison between the American and Chinese mathematics communities in the years immediately following the re-opening of China in 1978. Moreover, in the four decades from 1946 to 1984, he founded or co-founded three mathematics institutes; the one in the U.S., the Mathematical Sciences Research Institute, is thriving in Berkeley.
He was born on October 26, 1911, in Jiaxing, Zhejiang Province, China, sixteen days after the revolution that overthrew the Manchurian Dynasty and ushered in modern China. Typically for that era in China, his schooling was haphazard. He had one day of elementary education, four years of middle-high school, and at age fifteen skipped two grades to enter Nankai University. Under the tutelage of a great teacher in Nankai, he began to learn some substantive mathematics. To the end of his life, he retained a fond memory of his Nankai years, and this fact was to play a role in his eventual decision to return to China in 1999. His next four years (1930–34) were spent as a graduate student at Qing Hua University in Beijing (then Peiping), and he published three papers on the subject of projective differential geometry. While he sensed that his future lay in the general area of geometry, he reminisced in his later years that he was at a loss “how to climb this beautiful mountain.” In 1932, Wilhelm Blaschke of Hamburg, Germany, came to visit Beijing and gave some lectures on web geometry. Although those lectures were quite elementary, they opened his eyes to the fact that the era of projective differential geometry had come and gone, and something vaguely called “global differential geometry” was beckoning on the horizon. When he won a scholarship in 1934 to study abroad, he defied the conventional wisdom of going to the U.S. and chose to attend the University of Hamburg instead. This is the first of three major decisions he made in the ten-year period 1934–1943 that shaped the rest of his life. As we shall see, the decision in each case was by no means easy or obvious, but it only appeared to be so with hindsight. Throughout his life, he never seemed to lose this uncanny ability to make the right decision at the right time.
Soon after his arrival at Hamburg, he solved one of Blaschke’s problems in web geometry and was awarded a doctorate in 1936. However, the most important discovery he made during his two years in Hamburg was the work of Élie Cartan. The discovery was due to not only the fact that Blaschke was one of the few at the time who understood and recognized the importance of Cartan’s geometric work, but also the happy coincidence that E. Kähler had just published what we now call the Cartan–Kähler theory on exterior differential systems, and was giving a seminar on this theory at Hamburg. When Chern was given a postdoctoral fellowship in 1936 to pursue further study in Europe, he sought Blaschke’s advice. The latter presented him with two choices: either stay in Hamburg to learn algebraic number theory from Emil Artin, or go to Paris to learn geometry from Élie Cartan. At the time, Artin was a major star; he was also a phenomenal teacher, as Chern knew very well firsthand. But Chern made his second major decision by choosing Élie Cartan and Paris. His one-year stay in Paris (1936–37) was, in his own words, “unforgettable.” He got to know the master’s work directly from the master himself, and Cartan’s influence on his scientific outlook can be seen on almost every page of his four-volume Selected Papers (1978–1989).
Even before he returned to China in 1937, Chern had been appointed professor of mathematics at Qing Hua University, his former graduate school. Unfortunately, the Sino-Japanese War broke out in Northeast China when he was still in Paris, and Qing Hua University was moved to Kunming in southwestern China as part of the Southwest Associated University. It was to be ten more years before he could set his eyes again on the Qing Hua campus in Beijing. During 1937–43, he taught and studied in isolation in Kunming under harsh war conditions. It must be said that sometimes a little isolation is not a bad thing for people engaging in creative work. For Chern, those years broadened and deepened his understanding of Cartan’s work. He wrote near the end of his life that as a result of his isolation, he got to read over 70% of Cartan’s papers which total 4,750 pages. Another good thing that came out of those years was his marriage to Shih-ning Cheng in 1939, although a few months later, his pregnant bride had to leave him to return to Shanghai for reasons of personal safety. Their son Paul was born the following year but did not get to meet his father until he was six years old.
In all those years, he kept up his research and his papers appeared in international journals, including two in the Annals of Mathematics in 1942. Of the latter, the one on integral geometry [1] was reviewed in the Mathematical Reviews by André Weil who gave it high praise, and the other on isotropic surfaces [2] was refereed by none other than Hermann Weyl, who made this fact known to Chern himself when they finally met 1n 1943. Weyl read every line of the manuscript of [2], made suggestions for improvement, and recommended it with enthusiasm. But Chern was not satisfied with just being a known quantity to the mathematical elite because he wanted to find his own mathematical voice. When invitation to visit the Institute for Advanced Study (IAS) at Princeton came from O. Veblen and Weyl in 1943, he seized the opportunity and accepted in spite of the hardship of wartime travel. At considerable personal risk, he spent seven days to fly by military aircrafts from Kunming to Miami via India, Africa, and South America. He reached Princeton in August by train. The visit to IAS was his third major decision of the preceding decade, and perhaps the most important of all.
His sojourn at the IAS from August 1943 to December of 1945 changed
the course of differential geometry and transcendental algebraic
geometry; it changed his whole life as well.
Soon after his arrival at Princeton, he made a discovery that not only
solved one of the major problems of the
day — to
find an intrinsic proof of the
Let
The problem with the Allendoerfer–Weil proof is that it is conceptually
complex: as the phrase “Riemannian polyhedra” in the
title of
[e1]
suggests, it begins by triangulating
Consider the frame bundle
The last fact about
Chern’s proof of
This proof of
For the concluding step in the Chern proof of
We may interpret the preceding proof in the following way. The form
The whole idea of using the curvature form on a principal bundle to generate
characteristic classes is now so standard that it is difficult for us,
sixty years after the fact, to fully appreciate the startling originality
of Chern’s contribution. The fact revealed by
Les espaces fibrés … Leur rôle en géométrie différentielle, et tout particulièrement dans l’oeuvre d’Élie Cartan a été longtemps resté implicite, mais s’était clarifié peu à peu grâce aux travaux d’Ehresmann et surtout à ceux de Chern. La démonstration par Chern de la formule de Gauss–Bonnet et sa découverte des classes caractéristiques des variétés à structure complexe ou quasi-complexe avaient inauguré une nouvelle époque en géométrie différentielle, … [e8], p. 566.
[Chern and I] were both beginning to realize the major role which fibre bundles were playing, still mostly behind the scenes, in all kinds of geometric problems. … I will merely point out what can now be realized in retrospect about Chern’s proof for the Gauss–Bonnet theorem, as compared with the one Allendoerfer and I had given in 1942, following the footsteps of H. Weyl and other writers. The latter proof, resting on the consideration of “tubes,” did depend (although this was not apparent at the time) on the construction of a sphere-bundle, but of a non-intrinsic one, viz., the transversal bundle for a given immersion in Euclidean space; Chern’s proof operated explicitly for the first time with an intrinsic bundle, the bundle of tangent vectors of length 1, thus clarifying the whole subject once and for all. [e7], p. x–xi.
These passages may also shed some light on why Weil’s admiration of Chern never flagged throughout his life.
It was already mentioned that Chern began his quest for defining general
characteristic classes almost as soon as he saw how to prove the Gauss–Bonnet
theorem. To cut a long story short, the result of this work is the
substance of his paper
[5].
Briefly,
let a Hermitian metric be given on an
Now let
When the Hermitian metric is Kählerian, Chern identified the
To round off the picture, it should be pointed out that the analogue of the Chern forms for the orthogonal group was introduced around the same time by Pontryagin [e2], though the details came later [e3].
The topology of the forties was preoccupied with the real category, and Chern’s work on the characteristic classes of complex manifolds appeared at first to be slightly out of step with the times. But the dramatic growth of algebraic geometry, particularly transcendental algebraic geometry, beginning with the fifties made him a prophet. Chern classes are important in algebraic geometry for at least two reasons. One is that the Chern classes of algebraic varieties suggested that they might furnish a firm foundation for the (then) confusing plethora of algebraic-geometric invariants, and Hodge was among the first to push for this point of view [e4]. Chern himself made important contributions in this direction, but F. Hirzebruch’s work in the fifties capped this development and made this vision a reality [e5]. A second and perhaps more important reason is that, many by-now standard arguments in algebraic geometry (e.g., those using the Kodaira vanishing theorem or applications of Yau’s solution of the Calabi Conjecture) are simply not possible without the curvature representations of the Chern classes of a bundle.
Chern’s fame began to spread after 1944, though slowly, in the American mathematics community, and he was invited to give a one-hour address in the 1945 summer meeting of the American Mathematical Society. In reviewing the text of that address [4], Heinz Hopf wrote in Mathematical Reviews that Chern’s work had ushered in a new era in global differential geometry. Thereafter, the global study of manifolds became the main direction of geometric research. At age thirty-four, he had realized his youthful dream by scaling one of the highest peaks on that “beautiful mountain.”
In April of 1946, Chern returned to China and was immediately entrusted with the creation of a mathematics institute for Academia Sinica in Nanking. That he did, and became its de facto director (the official title was “Deputy Director”). We normally envision a “mathematics institute” to be a gathering of scholars to explore the frontiers of research, but China was not yet ready for that kind of institute for lack of a sufficient number of such Chinese mathematical scholars. Being a realist from beginning to end, Chern turned the institute into the only thing it could have been, namely, China’s first true graduate school in mathematics. He recruited a group of young people and personally took charge of their education by teaching them the fundamentals of modern mathematics. Many of this group subsequently became leaders of the next generation of Chinese mathematicians.
By late 1948, the political situation in China had become so unstable that Veblen and Weyl began to be concerned about Chern’s safety. With the help of R. Oppenheimer, then director of IAS, Chern and his family managed to land safely on U.S. soil on New Year’s Day of 1949. He was to be a member of IAS for the spring semester and, in the fall, take up a faculty position at the University of Chicago where he would stay until 1960. In 1950, he gave a one-hour address at the International Congress of Mathematicians (held in Cambridge, Massachusetts) on the differential geometry of fibre bundles. It was in the decade of the fifties that Chern classes began to force their way into most mathematicians’ consciousness, due in no small part to the spectacular advances in algebraic geometry made by Kodaira, Hirzebruch, and others.
In 1960, Chern accepted the offer to come to the University of California at Berkeley. Upon his arrival, he immediately attracted a group of young geometers, and Berkeley in the sixties and seventies became the de facto geometry center of the world. Although he officially retired in 1979, he remained active in Berkeley’s departmental affairs until the mid-eighties, and made Berkeley his home until 1999. Many honors came his way during the Berkeley years, the principal ones being the election to the National Academy of Sciences in 1961, the U.S. National Medal of Science in 1975, and the Wolf Prize from the Israel government in 1984. Later, he also received the Lobachevsky Prize from the Russian Academy in 2002, and the first Shaw Prize in mathematics in 2004, a few months before his death. In 2002, he was Honorary President of the International Congress of Mathematicians held at Beijing.
Chern’s leadership position in differential geometry was, if anything, enhanced
by his work in his Berkeley years. Two of his major papers in this period
hark back to his early work on characteristic classes. On the latter, he
was wont to point out that his main contribution to
characteristic classes was not so much the introduction of Chern classes as
the discovery of explicit
differential forms that represent those classes.
To him, it was the forms that give geometers an edge over topologists
in studying many aspects of these classes. With examples like
Yau’s
solution of the Calabi Conjecture in mind, one can hardly disagree
with him. The two pieces of work to be discussed
further justify his point of view.
In his collaboration with
Raoul Bott
[7]
in 1965
on generalized Nevanlinna theory in higher dimensions, they
constructed for the holomorphic category the “correct” version of
transgression
(cf.
The second paper related to Chern’s earlier work on
characteristic classes dates from 1971, when
he and
Jim Simons
introduced the Chern–Simons invariants
[9].
Let
So far, the forms
In case
The Chern–Simons invariants cannot be defined unless we have a Pontryagin form equal to 0. This naturally raises the question of whether on a given manifold with a vanishing Pontryagin class, there is a Riemannian metric whose corresponding Pontryagin form is zero.
One more major piece of work that Chern did in his Berkeley years should not go unmentioned. In 1974, he and Jürgen Moser wrote a paper in a completely different direction [8]. Generalizing Élie Cartan’s work on real hypersurfaces of complex Euclidean space of dimension two, they defined what we now call the Chern–Moser invariants of such hypersurfaces in all dimensions. These invariants are a complete set of local invariants in the real analytic case. The study of these invariants is now a fundamental part of geometric complex analysis. Finally, in 1992, when he was already eighty, he found inspiration in his own work in the late forties and, with D. Bao and Z. Shen, made a strong advocacy for generalizing classical Riemannian geometry to the Finsler setting. This advocacy has attracted a following.
During his Berkeley years, his leadership was felt in other areas too, but none more so than in the founding of two mathematics institutes. In 1981, the proposal he made jointly with Calvin Moore and I. M. Singer to establish an institute in mathematics on the Berkeley campus was officially approved by the government, and the Mathematics Sciences Research Institute (MSRI) was born. Chern served as its first director until 1984. The operational model of MSRI differs significantly from the most eminent research institute of our time, the Princeton Institute for Advanced Study. In contrast with the latter, MSRI has no permanent faculty, and each year its activities are organized around clearly defined mathematical topics. Senior mathematicians in each topic area are invited to visit MSRI for (part of) the year to help organize the scientific activities. This model has been followed around the world by other institutes since then.
Starting in the seventies, Chern took the lead in re-establishing mathematical communications between the U.S. and China. After his official retirement from the University in 1979, his visits to China became more frequent. Given that China has venerated scholarship for three thousand years, it was easy for someone with Chern’s diplomatic skills and preeminence to function smoothly at the highest political level in China. This may partially explain how he was able to establish, in 1984, a mathematics research institute in his alma mater, Nankai University, in Tianjin. A main goal of the Nankai Institute has been to attract leading mathematicians around the world to visit Tianjin and make it an active center of mathematics. Chern pursued this goal with vigor, and the Chinese government did its share in making foreign visitors welcome. When Chern finally returned to China for good in 1999, the well-being of the institute became his final project. He made ambitious plans that were only partially realized at the time of his death.
Chern is survived by his son Paul L. Chern, daughter May P. Chu, and four grandchildren, Melissa, Theresa, Claire, and Albert. His wife of sixty years, Shih-Ning, passed away earlier in year 2000 in Tianjin.