#### by Richard S. Palais and Chuu-Lian Terng

#### Introduction

Many mathematicians consider Shiing-Shen Chern to be the outstanding contributor to research in differential geometry in the second half of the twentieth century. Just as geometry in the first half-century bears the indelible stamp of Élie Cartan, so the seal of Chern appears large on the canvas of geometry that has been painted in the past fifty years. And beyond the great respect and admiration that his scientific accomplishments have brought him, there is also a remarkable affection and esteem for Chern on the part of countless colleagues, students, and personal friends. This reflects another aspect of his career — the friendship, warmth, and consideration Chern has always shown to others throughout a life devoted as much to helping younger mathematicians develop their full potential as to his own research.

Our recounting of Chern’s life is in two sections: the first, more biographical in nature, concentrates on details of his personal and family history; the second gives a brief report on his research and its influence on the development of twentieth-century mathematics.

Our main sources for the preparation of this article were the four volumes of Chern’s selected papers [24], [28], [30], [29] published by Springer-Verlag, a collection of Chern’s Chinese articles by Science Press [27], and many conversations with Chern himself.

#### Early life

Chern was born on October 28, 1911 in Jia Xin. His father, Bao Zheng Chern, passed the city level Civil Service examinations at the end of the Qing Dynasty, and later graduated from Zhe Jiang Law School and practiced law. He and Chern’s mother, Mei Han, had one other son and two daughters.

Because his grandmother liked to have him at home, Shiing-Shen was not sent to elementary school, but instead learned Chinese at home from his aunt. His father was often away working for the government, but once when his father was at home he taught Shiing-Shen about numbers, and the four arithmetic operations. After his father left, Shiing-Shen went on to teach himself arithmetic by working out many exercises in the three volumes of Bi Shuan Mathematics. Because of this he easily passed the examination and entered Xiu Zhou School, fifth grade, in 1920.

His father worked for the court in Tianjin and decided to move
the family there in 1922. Chern entered Fu Luen middle school that
year and continued to find mathematics easy and interesting. He
worked a large number of exercises in *Higher Algebra* by Hall and
Knight, and in *Geometry and Trigonometry* by Wentworth and Smith. He
also enjoyed reading and writing.

#### 1926–30, Nankai University

Chern passed the college entrance examinations in 1926, at age fifteen, and entered Nankai University to study Mathematics. In the late 1920s there were few mathematicians with a PhD degree in all of China, but Chern’s teacher, Lifu Jiang, had received a doctoral degree from Harvard with Julian Coolidge. Jiang had a strong influence on Chern’s course of study; he was very serious about his teaching, giving many exercises and personally correcting all of them. Nankai provided Chern with an excellent education during four happy years.

#### 1930–34, Qing Hua graduate school

In the early 1930s, many mathematicians with PhD degrees recently earned abroad were returning to China and starting to train students. It appeared to Chern that this new generation of teachers did not encourage students to become original and strike out on their own, but instead set them to work on problems that were fairly routine generalizations of their own thesis research. Chern realized that to attain his goal of high quality advanced training in mathematics he would have to study abroad. Since his family could not cover the expense this would involve, he knew that he would require the support of a government fellowship. He learned that a student graduating from Qing Hua graduate school with sufficiently distinguished records could be sent abroad with support for further study, so, after graduating from Nankai in 1930, he took and passed the entrance examination for Qing Hua graduate school. At that time the four professors of mathematics at Qing Hua were Qinglai Xiong, Guangyuan Sun, Wuzhi Yang (C. N. Yang’s father), and Zhifan Zheng (Chern’s father-in-law to be), and Chern studied projective differential geometry with Professor Sun.

While at Nankai Chern had taken courses from Jiang on the theory of curves and surfaces, using a textbook written by W. Blaschke. Chern had found this deep and fascinating, so when Blaschke visited Beijing in 1932, Chern attended all of his series of six lectures on web geometry. In 1934, when Chern graduated from Qing Hua, he was awarded a two-year fellowship for study in the United States but, because of his high regard for Blaschke, he requested permission from Qing Hua to use the fellowship at the University of Hamburg instead. The acting chairman, Professor Wuzhi Yang, helped both to arrange the fellowship for Chern and for his permission to use it in Germany. This was the year that the Nazis were starting to expel Jewish professors from the German universities, but Hamburg University had opened only several years before and, perhaps because it was so new, it remained relatively calm and a good place for a young mathematician to study.

#### 1934–36, Hamburg University

Chern arrived at Hamburg University in September of 1934, and started working under Blaschke’s direction on applications of Cartan’s methods in differential geometry. He received the Doctor of Science degree in February 1936. Because Blaschke travelled frequently, Chern worked much of the time with Blaschke’s assistant, Kähler. Perhaps a the major influence on him while at Hamburg was Kähler’s seminar on what is now a known as Cartan–Kähler Theory. This was then a new theory and everyone at the a Institute attended the first meeting. By the end of the seminar only Chern was left, but he felt that he had benefited greatly from it. When his two year fellowship ended in the summer of 1936, Chern was offered appointments at both Qing Hua and Beijing University. But he was also offered another year of support from The Chinese Culture Foundation and, with the recommendation of Blaschke, he went to Paris in 1936–37 to work under the renowned geometer Élie Cartan.

#### 1936–37, Paris

When Chern arrived in Paris in September of 1936, Cartan had so many students eager to work with him that they lined up to see him during his office hours. Fortunately, after two months Cartan invited Chern to see him at home for an hour once every other week during the remaining ten months he was in Paris. Chern spent all his efforts preparing for these biweekly meetings, working very hard and very happily. He learned moving frames, the method of equivalence, more of Cartan–Kähler theory, and most a importantly according to Chern himself, he learned the mathematical language and the way of thinking of Cartan. The three papers he wrote during this period represented the fruits of only a small part of the research that came out of this association with Cartan.

#### 1937–43, Kunming and the Southwest University Consortium

Chern received an appointment as Professor of Mathematics at Qing Hua in 1937. But before he could return to China, invading Japanese forces had touched off the long and tragic Sino-Japanese war. Qing Hua joined with Peking University and Nankai University to form a three-university consortium, first at Changsha, and then, beginning in January 1938, at Kunming, where it was called the Southwest Associated University. Chern taught at both places. It had an excellent faculty, and in particular Luogeng Hua was also Professor of Mathematics there. Chern had many excellent students in Kunming, some of whom later made substantial contributions to mathematics and physics. Among these were the mathematician H. C. Wang and the Nobel prize-winning physicist C. N. Yang. Because of the war, there was little communication with the outside world and the material life was meager. But Chern was fortunate enough to have Cartan’s recent papers to study, and he immersed himself in these and in his own research. The work begun during this difficult time would later become a major source of inspiration in modern mathematics.

#### Chern’s family

In 1937 Chern and Ms. Shih-Ning Cheng became engaged in Changsha, having been introduced by Wuzhi Yang. She had recently graduated from Dong Wu University, where she had studied biology. They were married in July of 1939, and Mrs. Chern went to Shanghai in 1940 to give birth to their first child, a son Buo Lung. The war separated the family for six years and they were not reunited until 1946. They have a second child, a daughter, Pu (married to Chingwu Chu, one of the main contributors in the development of high temperature superconductors).

The Cherns have had a beautiful and full marriage and family life. Mrs. Chern has always been at his side and Chern greatly appreciated her efforts to maintain a serene environment for his research. He expressed this in a poem he wrote on her sixtieth birthday:

Thirty-six years together

Through times of happiness

And times of worry too.

Time’s passage has no mercy.We fly the Skies and cross the Oceans

To fulfill my destiny;

Raising the children fell

Entirely on your shoulders.How fortunate I am

To have my works to look back upon,

I feel regrets you still have chores.Growing old together in El Cerrito is a blessing.

Time passes by,

And we hardly notice.

In 1978 Chern wrote in the article “A summary of my scientific life and works”:

“I would not conclude this account without mentioning my wife’s role in my life and work. Through war and peace and through bad and good times we have shared a life for forty years, which is both simple and rich. If there is credit for my mathematical works, it will be hers as well as mine.”

#### 1943–45, Institute for Advanced Study at Princeton

By now Chern was recognized as one of the outstanding mathematicians of China, and his work was drawing international attention. But he felt unsatisfied with his achievements, and when O. Veblen obtained a membership for him at the Institute for Advanced Study in 1943, he decided to go despite the great difficulties of wartime travel. In fact, it required seven days for Chern to reach the United States by military aircraft!

This was one of the most momentous decisions of
Chern’s life, for in those next two years in Princeton he was to
complete some of his most original and influential work.
In particular, he found an intrinsic proof of The Generalized
Gauss–Bonnet Theorem
[9],
and this in turn lead him to discover the
famous Chern characteristic classes
[10].
In 1945 Chern gave an
invited hour address to the American Mathematical Society, summarizing
some of these striking new advances. The written version of this talk
[11]
was an unusually influential paper, and as
Heinz Hopf
remarked in
reviewing it for *Mathematical Reviews* it signaled the arrival of a new
age in global differential geometry (“Dieser Vortrag… zeigt, dass
wir uns einer neuen Epoche in der ‘Differentialgeometrie im Grossen’
befinden”).

#### 1946–48, Academia Sinica

Chern returned to China in the spring of 1946. The Chinese government had just decided to set up an Institute of Mathematics as part of Academia Sinica. Lifu Jiang was designated chairman of the organizing committee, and he in turn appointed Chern as one of the committee members. Jiang himself soon went abroad, and the actual work of organizing the Institute fell to Chern. At the Institute, temporarily located in Shanghai, Chern emphasized the training of young people. He selected the best recent undergraduates from universities all over China and lectured to them twelve hours a week on recent advances in topology. Many of today’s outstanding Chinese mathematicians came from this group, including Wenjun Wu, Shantao Liao, Guo Tsai Chen, and C. T. Yang. In 1948 the Institute moved to Nanjing, and Academia Sinica elected eighty-one charter members, Chern being the youngest of these.

Chern was so involved in his research and with the training of students that he paid scant attention to the civil war that was engulfing China. One day however, he received a telegram from J. Robert Oppenheimer, then Director of the Institute for Advanced Study, saying “If there is anything we can do to facilitate your coming to this country please let us know.” Chern went to read the English language newspapers and, realizing that Nanjing would soon become embroiled in the turmoil that was rapidly overtaking the country, he decided to move the whole family to America. Shortly before leaving China he was also offered a position at the Tata Institute in Bombay. The Cherns left from Shanghai on December 31, 1948, and spent the Spring Semester at the Institute in Princeton.

#### 1949–60, Chicago University

Chern quickly realized that he would not soon be able to return to China, and so would have to find a permanent position abroad. At this time, Professor Marshall Stone of the University of Chicago Mathematics Department had embarked on an aggressive program of bringing to Chicago stellar research figures from all over the world, and in a few years time he had made the Chicago department one of the premier centers for mathematical research and graduate education worldwide. Among this group of outstanding scholars was Chern’s old friend, André Weil, and in the summer of 1949 Chern too accepted a professorship at the University of Chicago. During his eleven years there Chern had ten doctoral students. He left in 1960 for the University of California at Berkeley, where he remained until his retirement in 1979.

#### Chern and C. N. Yang

Chern’s paper on characteristic classes was published in 1946 and he gave a one semester course on the theory of connections in 1949. Yang and Mills published their paper introducing the Yang–Mills theory into physics in 1954. Chern and Yang were together in Chicago in 1949 and again in Princeton in 1954. They are good friends and often met and discussed their respective research. Remarkably, neither realized until many years later that they had been studying different aspects of the same thing!

#### 1960–79, UC Berkeley

Chern has commented that two factors convinced him to make the move to Berkeley. One was that the Berkeley department was growing vigorously, giving him the opportunity to build a strong group in geometry. The other was… the warm weather. During his years at Berkeley, Chern directed the thesis research of thirty-one students. He was also teacher and mentor to many of the young postdoctoral mathematicians who came to Berkeley for their first jobs. (This group includes one of the coauthors of this article; the other was similarly privileged at Chicago.) During this period the Berkeley Department became a world-famous center for research in geometry and topology. Almost all geometers in the United States, and in much of the rest of the world too, have met Chern and been strongly influenced by him. He has always been friendly, encouraging, and easy to talk with on a personal level, and since the 1950s his research papers, lecture notes, and monographs have been the standard source for students desiring to learn differential geometry. When he “retired” from Berkeley in 1979, there was a week long “Chern Symposium” in his honor, attended by over three hundred geometers. In reality, this was a retirement in name only; during the five years that followed, not only did Chern find time to continue occasional teaching as Professor Emeritus, but he also went “up the hill” to serve as the founding director of the Berkeley Mathematical Sciences Research Institute (MSRI).

#### 1981–present, the three institutes

In 1981 Chern, together with Calvin Moore, Isadore Singer, and several other San Francisco Bay area mathematicians wrote a proposal to the National Science Foundation for a mathematical research institute at Berkeley. Of the many such proposals submitted, this was one of only two that were eventually funded by the NSF. Chern became the first director of the resulting Mathematical Sciences Research Institute (MSRI), serving in this capacity until 1984. MSRI quickly became a highly successful institute and many credit Chern’s influence as a major factor.

In fact, Chern has been instrumental in establishing three important institutes of mathematical research: The Mathematical Institute of Academia Sinica (1946), The Mathematical Sciences Research Institute in Berkeley, California (1981), and The Nankai Institute for Mathematics in Tianjin, China (1985). It was remarkable that Chern did this despite a reluctance to get involved with details of administration. In such matters his adoption of Laozi’s philosophy of “Wu Wei” (roughly translated as “Let nature take its course”) seems to have worked admirably. Chern has always believed strongly that China could and should become a world leader in mathematics. But for this to happen he felt two preconditions were required:

The existence within the Chinese mathematical community of a group of strong, confident, creative people, who are dedicated, unselfish, and aspire to go beyond their teachers, even as they wish their students to go beyond them.

Ample support for excellent library facilities, research space, and communication with the world-wide mathematical community. (Chern claimed that these resources were as essential for mathematics as laboratories were for the experimental sciences).

It was to help in achieving these goals that Chern accepted the job of organizing the mathematics institute of Academia Sinica during 1946 to 1948, and the reason why he returned to Tianjin to found the Mathematics Institute at Nankai University after his retirement in 1984 as director of MSRI.

During 1965–76, because of the Cultural Revolution, China lost a whole generation of mathematicians, and with them much of the tradition of mathematical research. Chern started visiting China frequently after 1972, to lecture, to train Chinese mathematicians, and to rekindle these traditions. In part because of the strong bonds he had with Nankai University, he founded the Nankai Mathematical Research Institute there in 1985. This Institute has its own housing, and attracts many visitors both from China and abroad. In some ways it is modeled after the Institute for Advanced Study in Princeton. One of its purposes is to have a place where mature mathematicians and graduate students from all of China can spend a period of time in contact with each other and with foreign mathematicians, concentrating fully on research. Another is to have an inspiring place in which to work; one that will be an incentive for the very best young mathematicians who get their doctoral degrees abroad to return home to China.

#### Honors and awards

Chern was invited three times to address The International Congress of Mathematicians. He gave an Hour Address at the 1950 Congress in Cambridge, Massachusetts (the first ICM following the Second World War), spoke again in 1958, at Edinburgh, Scotland, and was invited to give a second Hour Address at the 1970 ICM in Nice, France. These Congresses are held only every fourth year and it is unusual for a mathematician to be invited twice to give a plenary Hour Address.

During his long career Chern was awarded numerous honorary degrees. He was elected to the US National Academy of Sciences in 1961, and received the National Medal of Science in 1975 and the Wolf Prize in 1983. The Wolf Prize was instituted in 1979 by the Wolf Foundation of Israel to honor scientists who had made outstanding contributions to their field of research. Chern donated the prize money he received from this award to the Nankai Mathematical Institute. He is also a foreign member of The Royal Society of London, Academie Lincei, and the French Academy of Sciences. A more complete list of the honors he received can be found in the Curriculum Vitae in [28].

#### An overview of Chern’s research

Chern’s mathematical interests have been unusually wide and far-ranging and he has made significant contributions to many areas of geometry, both classical and modern. Principal among these are:

Geometric structures and their equivalence problems

Integral geometry

Euclidean differential geometry

Minimal surfaces and minimal submanifolds

Holomorphic maps

Webs

Exterior Differential Systems and Partial Differential Equations

The Gauss–Bonnet Theorem

Characteristic classes

Since it would be impossible within the space
at our disposal to present a detailed review of Chern’s achievements
in so many areas, rather than attempting a superficial account of all
facets of his research, we have elected to concentrate on those areas
where the effects of his contributions have, in our opinion, been most
profound and far-reaching. For further information concerning Chern’s
scientific contributions the reader may consult the four volume set,
*Shiing-Shen Chern: Selected Papers*
[24],
[28],
[30],
[29].
This includes a Curriculum
Vitae, a full bibliography of his published papers, articles of
commentary by André Weil and
Phillip Griffiths,
and a scientific
autobiography in which Chern comments briefly on many of his papers.

One further *caveat*; the reader should keep in mind that this is a
mathematical biography, *not* a mathematical history. As such, it
concentrates on giving an account of Chern’s own scientific
contributions, mentioning other mathematicians only if they were his
coauthors or had some particularly direct and personal effect on
Chern’s research. Chern was working at the cutting edge of mathematics
and there were of course many occasions when others made discoveries
closely related to Chern’s and at approximately the same time. A far
longer (and different) article would have been required if we had even
attempted to analyze such cases. But it is not only for reasons of
space that we have avoided these issues. A full historical treatment
covering this same ground would be an extremely valuable undertaking,
and will no doubt one day be written. But that will require a major
research effort of a kind that neither of the present authors has the
training or qualifications even to attempt.

Before turning to a description of Chern’s research, we would like to point out a unifying theme that runs through all of it: his absolute mastery of the techniques of differential forms and his artful application of these techniques in solving geometric problems. This was a magic mantle, handed down to him by his great teacher, Élie Cartan. It permitted him to explore in depth new mathematical territory where others could not enter. What makes differential forms such an ideal tool for studying local and global geometric properties (and for relating them to each other) is their two complementary aspects. They admit, on the one hand, the local operation of exterior differentiation, and on the other the global operation of integration over cochains, and these are related via Stokes’ Theorem.

#### Geometric structures and their equivalence problems

Much of Chern’s early work was concerned with various “equivalence
problems”. Basically, the question is how to determine effectively
when two geometric structures of the same type are “equivalent” under
an appropriate group of geometric transformations. For example, given
two curves in space, when is there a Euclidean motion that carries one
onto the other? Similarly, when are two Riemannian structures locally
isometric? Classically one tried to associate with a given type of
geometric structure various “invariants”, that is, simpler and
better understood objects that do not change under an isomorphism, and
then show that certain of these invariants are a “complete set”, in
the sense that they determine the structure up to isomorphism. Ideally
one should also be able to specify what values these invariants can
assume by giving relations between them that are both necessary and
sufficient for the existence of a structure with a given set of
invariants. The goal is a theorem like the elegant classic paradigm of
Euclidean plane geometry, stating that the three side lengths of a
triangle determine it up to congruence, and that three positive real
numbers arise as side lengths precisely when each is less than the sum
of the other two. For smooth, regular space curves the solution to the
equivalence problem was known early in the last century. If to a given
space curve __\( \sigma(s) \)__ (parametrized by arc length) we associate its
curvature __\( \kappa(s) \)__ and torsion __\( \tau(s) \)__, it is easy to show that these two
smooth scalar functions are invariant under the group of Euclidean
motions, and that they uniquely determine a curve up to an element of
that group. Moreover any smooth real valued functions __\( \kappa \)__ and __\( \tau \)__ can
serve as curvature and torsion as long as __\( \kappa \)__ is positive. The more
complex equivalence problem for surfaces in space had also been solved
by the mid 1800s. Here the invariants turned out to be two smooth
quadratic forms on the surface, the first and second fundamental
forms, of which the first, the metric tensor, had to be positive
definite and the two had to satisfy the so-called Gauss and Codazzi
equations. The so-called “form problem”, that is the local equivalence
problem for Riemannian metrics, was also solved classically (by
Christoffel and
Lipschitz). The solution is still more complex and
superficially seems to have little in common with the other examples
above.

As Chern was starting his research career, a major challenge facing
geometry was
to find what this seemingly disparate class of examples had in common,
and thereby
discover a general framework for the Equivalence Problem. Cartan saw
this clearly,
and had already made important steps in that direction with his
general machinery
of “moving frames”. His approach was to reduce a general equivalence
problem to
one of a special class of equivalence problems for differential forms.
More precisely, he
would associate to a given type of local geometric structure in open
sets __\( U \)__ of __\( \mathbf{R}^n \)__, an
“equivalent” structure, given by specifying:

a subgroup

__\( G \)__of__\( \mathbf{GL}(n, \mathbf{R}) \)__,certain local coframe fields

__\( \{\theta_i \} \)__in open subsets__\( U \)__of__\( \mathbf{R}^n \)__(i.e.,__\( n \)__linearly independent differential 1-forms in__\( U \)__).

The condition of equivalence for __\( \{\theta_i\} \)__ in __\( U \)__ and
__\( \{\theta_i^{\ast}\} \)__ in __\( U^{\ast} \)__ is the existence of a
diffeomorphism __\( \varphi \)__ of __\( U \)__ with __\( U^{\ast} \)__ such that
__\[ \varphi^{\ast} (\theta_i^{\ast}) = \sum_{i=1}^n a_{ij} \theta_j ,\]__
where __\( (a_{ij} ) \)__ is a smooth map of __\( U \)__ into __\( G \)__. A geometric
structure defined by the choices (1) and (2) is now usually called a
“__\( G \)__-structure”, a name introduced by Chern in the course of
formalizing and explicating Cartan’s approach. For a given geometric
structure one must choose the related __\( G \)__-structure so that its notion
of equivalence coincides with that for the originally given geometric
structure, so the invariants of the __\( G \)__-structure will also be the
same as for the given geometric structure. In the case of the form
problem one takes __\( G = \mathbf{O}(n) \)__, and given a Riemannian
metric __\( ds^2 \)__ in __\( U \)__ chooses any __\( \theta_i \)__ such that
__\[ ds^2 = \sum_{i=1}^n\theta_i^2 \]__
in __\( U \)__. While not always so obvious as in
this case (and a real geometric insight is sometimes required for
their discovery) most other natural geometric equivalence problems,
including the ones mentioned above, do admit reformulation in terms of
__\( G \)__-structures.

But do we gain anything besides uniformity from such a reformulation?
In fact, we do, for Cartan also developed general techniques for
finding complete sets of invariants for __\( G \)__-structures. Unfortunately,
however, carrying out this solution of the Equivalence Problem in
complete generality depends on his powerful but difficult theory of
Pfaffian systems in involution, with its method of prolongation, a
theory not widely known or well understood even today. In fact, while
his preeminence as a geometer was clearly recognized towards the end
of his career, many great mathematicians confessed to finding Cartan’s
work hard going at best, and few mathematicians of his day were able
to comprehend fully his more novel and innovative advances. For
example, in a review of one of his books (*Bull. Amer. Math. Soc.*
vol. 44, p. 601)
H. Weyl
made this often quoted admission:

“Cartan is undoubtedly the greatest living master in differential geometry… Nevertheless I must admit that I found the book, like most of Cartan’s papers, hard reading…”

Given this well-known difficulty Cartan had in communicating his more esoteric ideas, one can easily imagine that his important insights on the Equivalence Problem might have lain buried. Fortunately they were spared such a fate.

Recall that Chern had spent his time at Hamburg
studying the Cartan-Kähler a theory of Pfaffian systems with Kähler,
and immediately after Hamburg Chern spent a year in Paris continuing
his study of these techniques with Cartan. Clearly Chern was ideally
prepared to carry forward the attack on the Equivalence Problem. In a
series of beautiful papers over the next twenty years not only did he
do just that, but he also explained and reformulated the theory with
such clarity and geometric appeal that much (though by no means all!)
of the theory has become part of the common world-view of differential
geometers, to be found in the standard textbooks on geometry. Those
two decades were also, not coincidentally, the years that saw the
development of the theory of fiber bundles and of connections on
principal __\( G \)__-bundles. These theories were the result of the combined
research efforts of many people and had multiple sources of
inspiration both in topology and geometry. One major thread in that
development was Chern’s work on the Equivalence Problem and his
related research on characteristic classes that grew out of it. In
order to discuss this important work of Chern we must first define
some of the concepts and notations that he and others introduced.

Using current geometric terminology, a __\( G \)__-structure for a smooth
n-dimensional manifold __\( M \)__ is a reduction of the structure group of
its principal tangent coframe bundle from __\( \mathbf{GL}(n,
\mathbf{R}) \)__ to the subgroup __\( G \)__. In particular, the total space
of this reduction is a principal __\( G \)__-bundle, __\( P \)__, over __\( M \)__
consisting of the admissible coframes
__\[ \theta = (\theta_1, \dots, \theta_n) ,\]__
and we can identify the __\( G \)__-structure with this __\( P \)__. There
are __\( n \)__ canonically defined 1-forms __\( \omega_i \)__ on __\( P \)__; if
__\[ \Pi : P\rightarrow M \]__
is the bundle projection, then the value of __\( \omega_i \)__
at __\( \theta \)__ is __\( \Pi^{\ast} (\theta_i ) \)__. The kernel of __\( D\Pi \)__ is
of course the subbundle of the tangent bundle __\( T\!P \)__ of __\( P \)__ tangent to
its fibers, and is usually called the vertical subbundle __\( V \)__. Clearly
the canonical forms __\( \omega_i \)__ vanish on __\( V \)__. The group __\( G \)__ acts on
the right on __\( P \)__, acting simply transitively on each fiber, so we can
identify the vertical space __\( V_{\theta} \)__ at any point __\( \theta \)__ with
the Lie algebra __\( L(G) \)__ of left-invariant vector fields on __\( G \)__. Now, as
Ehresmann
first noted, a “connection” in Cartan’s sense for the
given __\( G \)__-structure (or as we now say, a __\( G \)__-connection for the
principal bundle __\( P \)__) is the same as a “horizontal” subbundle __\( H \)__
of __\( T\!P \)__ complementary to __\( V \)__ and invariant under __\( G \)__. Instead of __\( H \)__
it is equivalent to consider the projection of __\( T\!P \)__ onto __\( V \)__ along __\( H \)__
which, by the above identification of __\( V_{\theta} \)__ with __\( L(G) \)__, is an
__\( L(G) \)__-valued 1-form __\( \omega \)__ on __\( P \)__, called the “connection
1-form”. If we denote the right action of __\( g \in G \)__ on __\( P \)__ by __\( R_g \)__,
then the invariance of __\( H \)__ under __\( G \)__ translates to the transformation
law
__\[ R^{\ast}_g (\omega) = \operatorname{ad}(g^{-1}) \circ \omega \]__
for __\( \omega \)__,
where __\( \operatorname{ad} \)__ denotes the adjoint representation of __\( G \)__ on __\( L(G) \)__.
__\( L(G) \)__-valued forms on __\( P \)__ transforming in this way are called
equivariant. Since __\( L(G) \)__ is a subalgebra of the Lie algebra
__\( L(\mathbf{GL}(n, \mathbf{R})) \)__ of __\( n{\times}n \)__ matrices, we
can regard __\( \omega \)__ as an __\( n{\times}n \)__ matrix-valued 1-form on __\( P \)__, or
equivalently as a matrix __\( \omega_{ij} \)__ of __\( n^2 \)__ real-valued 1-forms on
__\( P \)__.

If __\( \sigma : [0, 1] \rightarrow M \)__ is a smooth path in __\( M \)__ from __\( p \)__ to
__\( q \)__, then the connection defines a canonical __\( G \)__-equivariant map
__\( \pi_{\sigma} \)__ of the fiber __\( P_p \)__ to the fiber __\( P_q \)__, called parallel
translation along __\( \sigma \)__; namely __\( \pi_{\sigma} (\theta)
=\tilde{\sigma}(1) \)__, where __\( \tilde{\sigma} \)__ is the unique horizontal
lift of __\( \sigma \)__ starting at __\( \theta \)__. In general, parallel
translation depends on the path __\( \sigma \)__, not just on the endpoints __\( p \)__
and __\( q \)__. If it depends only on the homotopy class of __\( \sigma \)__ with fixed
endpoints, then the connection is called “flat”. It is easy to see
that this is so if and only if the horizontal subbundle __\( H \)__ of __\( T\!P \)__
is integrable, and using the Frobenius integrability criterion, this
translates to
__\[ d\omega_{ij} =\sum_k \omega_{ik} \wedge \omega_{kj} .\]__
Thus it is natural to define the matrix __\( \Omega_{ij} \)__ of so-called
curvature forms of the connection, (whose vanishing is necessary and
sufficient for flatness) by
__\[ d\omega_{ij}= \sum_k \omega_{ik} \wedge
\omega_{kj} - \Omega_{ij} \]__
or, in matrix notation,
__\[ d\omega = \omega \wedge \omega - \Omega .\]__
Since __\( \omega \)__ is equivariant, so is
__\( \Omega \)__. Differentiating the defining equation of the curvature forms
gives the Bianchi identity,
__\[ d\Omega = \Omega \wedge \omega - \omega \wedge \Omega .\]__
A local cross-section __\( \theta : U \rightarrow P \)__ is
called an “admissible local coframe” for the __\( G \)__-structure, and we
can use it to pull back the connection forms and curvature forms
to forms __\( \psi_{ij} \)__ and __\( \Psi_{ij} \)__ on __\( U \)__. Any other admissible
coframe field __\( \hat{\theta} \)__ in __\( U \)__ is related to __\( \theta \)__ by a
unique “change of gauge”, __\( g \)__ in __\( U \)__ (i.e., a unique map __\( g : U
\rightarrow G \)__) such that
__\[ \hat{\theta}(x) = R_{g(x)} \theta(x) .\]__
If
we use __\( \hat{\theta} \)__ to also pull back the connection and curvature
forms to forms __\( \hat{\psi} \)__ and __\( \hat{\Psi} \)__ on __\( U \)__, then, using
matrix notation, it follows easily from the equivariance of __\( \omega \)__
and __\( \Omega \)__ that
__\[ \hat{\psi} = dg\, g^{-1} + g\psi g^{-1}
\quad\text{and}\quad
\hat{\Psi} = g\Psi g^{-1} .\]__

But where do connections fit into the Equivalence Problem? While
Cartan’s solution to the equivalence problem for __\( G \)__-structures was
complicated in the general case, it became much simpler for the
special case that __\( G \)__ is the trivial subgroup __\( \{e\} \)__. For this reason
Cartan had developed a method by which one could sometimes reduce a
__\( G \)__-structure on a manifold __\( M \)__ to an __\( \{e\} \)__-structure on a new
manifold obtained by “adding variables” corresponding to coordinates
in the group __\( G \)__. Chern recognized that this new manifold was just the
total space __\( P \)__ of the principal __\( G \)__-bundle, and that Cartan’s reduction
method amounted to finding an “intrinsic __\( G \)__-connection” for __\( P \)__, i.e.,
one canonically associated to the __\( G \)__-structure. Indeed the canonical
1-forms __\( \omega_i \)__ together with a linearly independent set of the connection
forms __\( \omega_{ij} \)__, defined by the intrinsic connection, give a canonical
coframe field for __\( P \)__, which of course is the same as an
__\( \{e\} \)__-structure. Finally, Chern realized that in this setting one could
describe geometrically the invariants for a __\( G \)__-structure given by
Cartan’s general method; in fact they can all be calculated from the
curvature forms of the intrinsic connection.

Note that this covers one of the most important examples of a
__\( G \)__-structure; namely the case __\( G = \mathbf{O}(n) \)__, corresponding to Riemannian
geometry. The intrinsic connection is of course the “Levi-Civita
connection”. Moreover, in this case it is also easy to explain how to
go on to “solve the form problem”, i.e., to find explicitly a complete
set of local invariants for a Riemannian metric. In fact, they can be
taken as the components of the Riemann curvature tensor and its
covariant derivatives in Riemannian normal coordinates. To see this,
note first the obvious fact that there is a local isometry of the
Riemannian manifold __\( (M, g) \)__ with __\( (M^{\ast}, g^{\ast}) \)__ carrying the orthonormal
frame __\( e_i \)__ at __\( p \)__ to __\( e^{\ast}_i \)__ at __\( p^{\ast} \)__ if and only if in some neighborhood of the
origin the components __\( g_{ij} (x) \)__ of the metric tensor of __\( M \)__ with respect
to the Riemannian normal coordinates __\( x_k \)__ defined by __\( e_i \)__ are *identical*
to the corresponding components __\( g^{\ast}_{ij} (x) \)__ of the metric tensor of __\( M^{\ast} \)__
with respect to Riemannian normal coordinates defined by __\( e^{\ast}_i \)__. The
proof is then completed by using the easy, classical fact
([e1],
Appendix II)
that each coefficient in the Maclaurin expansion of __\( g_{ij}
(x) \)__ can be expressed as a universal polynomial in the components of
the Riemann tensor and a finite number of its covariant derivatives.

Let us denote by __\( N (G) \)__ the semidirect product __\( G \ltimes
\mathbf{R}^n \)__ of affine transformations of __\( \mathbf{R}^n \)__
generated by __\( G \)__ and the translations. Correspondingly we can
“extend” the principal __\( G \)__-bundle __\( P \)__ of linear frames to the
associated principal __\( N (G) \)__-bundle __\( N (P ) \)__ of affine frames. Chern
noted in
[12]
that the above technique could be expressed more
naturally, and could be generalized to a wide class of groups __\( G \)__, if
one looked for intrinsic __\( N (G) \)__-connections on __\( N (P ) \)__. The
curvature of an __\( N (G) \)__-connection on __\( N (P ) \)__ is a two-form __\( \Omega \)__
on __\( N (P ) \)__ with values in the Lie algebra __\( L(N (G)) \)__ of __\( N (G) \)__. Now
__\( L(N (G)) \)__ splits canonically as the direct sum of __\( L(G) \)__ and
__\( L(\mathbf{R}^n) = \mathbf{R}^n \)__, and __\( \Omega \)__ splits
accordingly. The __\( \mathbf{R}^n \)__ valued part, __\( \tau \)__, of __\( \Omega \)__
is called the torsion of the connection, and what Chern exploited was
the fact that he could in certain cases define “intrinsic” __\( N (G) \)__
connections by putting conditions on __\( \tau \)__. For example, the
Levi-Civita connection can be characterized as the unique __\( N
(\mathbf{O}(n)) \)__ connection on __\( N (P ) \)__ such that __\( \tau = 0 \)__. In
fact, in
[12]
Chern showed that if the Lie algebra __\( L(G) \)__ satisfied a
certain simple algebraic condition (“property C”) then it was always
possible to define an intrinsic __\( N (G) \)__ connection in this way, and he
proved that any compact __\( G \)__ satisfies property C. He also pointed out
here, from the point of view of Cartan’s theory of pseudogroups, why
some __\( G \)__-structures do *not* admit intrinsic connections. The
pseudogroup of a __\( G \)__-structure __\( \Pi : P \rightarrow M \)__ is the
pseudogroup of local diffeomorphisms of __\( M \)__ whose differential
preserves the subbundle __\( P \)__. It is elementary that the group of bundle
automorphisms of a principal __\( G \)__-bundle that preserve a given
__\( G \)__-connection is a finite dimensional Lie group and so
*a fortiori* the pseudogroup of a __\( G \)__-structure with a
canonically defined connection will be a Lie group. But there are
important examples of groups __\( G \)__ for which the pseudogroup of a
__\( G \)__-structure is of infinite dimension. For example, if __\( n = 2m \)__ and
we take __\( G = \mathbf{GL}(m, \mathbf{C}) \)__, then a __\( G \)__-structure
is the same thing as an almost-complex structure, and the group of
automorphisms is an infinite pseudogroup.

Chern solved many concrete equivalence problems. In
[1]
and
[4]
he carried this out for
the path geometry defined by a third order ordinary differential
equation. Here the __\( G \)__-structure is on the contact manifold of unit
tangent vectors of __\( \mathbf{R}^2 \)__, and __\( G \)__ is the
ten-dimensional group of circle preserving contact transformations.
In
[2],
[3]
he generalized this to the
path geometry of systems of __\( n \)__-th order ordinary differential
equations. In
[8]
he considers a generalized
projective geometry, i.e., the geometry of __\( (k + 1)(n -
k) \)__-parameter family of __\( k \)__-dimensional submanifolds in
__\( \mathbf{R}^n \)__, and in
[6],
[7]
the
geometry defined by an __\( (n - 1) \)__-parameter family of hypersurfaces
in __\( \mathbf{R}^n \)__. In
[22]
(jointly with Moser) and in
[23]
he considers real hypersurfaces in __\( \mathbf{C}^n \)__.
This latter research played a fundamental role in the development
of the theory of CR manifolds.

#### Integral geometry

The group __\( G \)__ of rigid motions of __\( \mathbf{R}^n \)__ acts transitively on various spaces
__\( S \)__ of geometric objects (e.g., points, lines, affine subspaces of a
fixed dimension, spheres of a fixed radius) so that these spaces can
be regarded as homogeneous spaces, __\( G/H \)__, and the invariant measure on __\( G \)__
induces an invariant measure on __\( S \)__. This is the so-called “kinematic
density”, first introduced by Poincaré, and the basic problem of
integral geometry is to express the integrals of various
geometrically interesting quantities with respect to the kinematic
density in terms of known integral invariants (see
[17]).
The simplest
example is Crofton’s formula for a plane curve __\( C \)__,
__\[
\int n (\ell \cap C)\, d\ell= 2L (C)
\]__
where __\( L(C) \)__ is its length, __\( n(\ell \cap C) \)__ is the number of its
intersection points with a line in the plane, and __\( d\ell \)__ is the
kinematic density on lines. We can interpret this formula as saying
that the average number of times that a line meets a curve (i.e., is
incident with a point on the curve) is equal to twice the length of
the curve.

In [5], Chern laid down the foundations for a much generalized integral geometry. In [e4], André Weil says of this paper that:

“… it lifted the whole subject at one stroke to a higher plane than where Blaschke’s school had left it, and I was impressed by the unusual talent and depth of understanding that shone through it.”

Chern first extended the classical notion of “incidence” to a pair
of elements from two homogeneous spaces __\( G/H \)__ and __\( G/K \)__ of the same
group __\( G \)__. Given __\( aH \in G/H \)__ and __\( bK \in G/K \)__, Chern calls them
“incident” if
__\[ aH \cap bK \neq\emptyset .\]__
This definition plays an
important role in the theory of Tits buildings.

In
[13]
and
[17]
Chern obtained fundamental kinematic formulas for two
submanifolds in __\( \mathbf{R}^n \)__. The integral invariants in Chern’s formula
arise naturally in Weyl’s formula for the volume of a tube __\( T_{\rho} \)__ of
radius __\( \rho \)__ about a __\( k \)__-dimensional submanifold __\( X \)__ of
__\( \mathbf{R}^n \)__. Setting
__\( m = n - k \)__, Weyl’s formula is:
__\[
V (T_\rho) =\sum_{0\leq i\leq k,\, i \text{ even}} c_i \mu_i (X)\rho^{m+i}.
\]__
Here the __\( c_i \)__ are constants depending on __\( m \)__ and __\( i \)__,
__\[
\mu_i(X) = \int_M I_i(\Omega)
\]__
where __\( I_i \)__ is a certain adjoint invariant polynomial of degree __\( i/2 \)__
on the Lie algebra of __\( \mathbf{O}(n) \)__, and __\( \Omega \)__ is the
curvature form with respect to the induced metric on __\( X \)__. Chern’s
formula (also discovered independently by
Federer) is:
__\[
\int \mu_e (M_1 \cap g M_2) \, dg = \sum_{0 \leq i \leq e,\, i \text{
even}} c_i \mu_i (M_1) \mu_{e-i} (M_2),
\]__
where __\( M_1 \)__ and __\( M_2 \)__ are submanifolds of __\( \mathbf{R}^n \)__ of dimensions __\( p \)__ and __\( q \)__
respectively, __\( e \)__ is even, __\( 0 \leq e \)__ __\( \leq p + q - n \)__, and __\( c_i \)__ are constant depend
on __\( n, p, q, e \)__. Griffiths made the following comment concerning this
paper
[e3]:

“Chern’s proof of [this formula] exhibits a number of characteristic features. Of course, one is the use of moving frames…. Another is that the proof proceeds by direct computation rather than by establishing an elaborate, conceptual framework; in fact upon closer inspection there is such a conceptual framework, as described in [5], however, the philosophical basis is not isolated but is left to the reader to understand by seeing how it operates in a nontrivial problem.”

#### Euclidean differential geometry

One of the main topics in classical differential geometry is the study
of local invariants of submanifolds in Euclidean space under the group
of rigid motions, i.e., the equivalence problem for submanifolds. The
solution is classical. In fact, the first and second fundamental
forms, __\( I \)__ and __\( \mathit{II} \)__, and the induced connection __\( \nabla^\nu \)__ on the normal bundle of
a submanifold satisfy the Gauss, Codazzi and Ricci equations, and they
form a complete set of local invariants for submanifolds in __\( \mathbf{R}^n \)__.
Explicitly these invariants are as follows:

__\( I \)__is the induced metric on__\( M \)__,__\( \mathit{II} \)__is a quadratic form on__\( M \)__with values in the normal bundle__\( \nu(M ) \)__such that, for any unit tangent vector__\( u \)__and unit normal vector__\( v \)__at__\( p \)__,__\[ {\mathit{II}}_v (u) = \langle{\mathit{II}}(u), v\rangle \]__is the curvature at__\( p \)__of the plane curve__\( \sigma \)__formed by intersecting__\( M \)__with the plane spanned by__\( u \)__and__\( v \)__, andif

__\( s \)__is a smooth normal field then__\( \nabla^{\nu} (s) \)__is the orthogonal projection of the differential__\( ds \)__onto the normal bundle__\( \nu(M) \)__.

__\( {\mathit{II}}_v =
\langle{\mathit{II}},v\rangle \)__
is called the second fundamental form in the direction of
__\( v \)__. The self-adjoint operator __\( A_v \)__ corresponding to
__\( {\mathit{II}}_v \)__
is called the shape operator of __\( M \)__ in the
direction __\( v \)__.

Chern’s work in this field involved mainly the relation between the global geometry of submanifolds and these local invariants. He wrote many important papers in the area, but because of space limitations we will concentrate only on the following:

##### (1) Minimal surfaces

Since the first variation for the area functional for submanifolds of
__\( \mathbf{R}^n \)__ is the trace of the second fundamental form, a
submanifold __\( M \)__ of __\( \mathbf{R}^n \)__ is called minimal if
__\[ \operatorname{trace}(\mathit{II}\,) = 0 .\]__
Let __\( \mathbf{Gr}(2, n) \)__ denote the
Grassmann manifold of 2-planes in __\( \mathbf{R}^n \)__. The Gauss map
__\( G \)__ of a surface __\( M \)__ in __\( \mathbf{R}^n \)__ is the map from __\( M \)__ to the
Grassmann manifold __\( \mathbf{Gr}(2, n) \)__ defined by
__\[ G(x) = \text{the tangent plane to }M\text{ at }x .\]__
The Grassmann manifold
__\( \mathbf{Gr}(2, n) \)__ can be identified as the hyperquadric
__\[ z^2_1 + \dots + z^2_n = 0 \]__
of __\( \mathbf{{CP}}^{n-1} \)__ (via the map that sends
a 2-plane __\( V \)__ of __\( \mathbf{R}^n \)__ to the complex line spanned by __\( e_1 + ie_2 \)__,
where __\( (e_1 , e_2 ) \)__ is an orthonormal base for __\( V \)__). Thus
__\( \mathbf{Gr}(2, n) \)__ has a complex structure. On the other hand, an
oriented surface in __\( \mathbf{R}^n \)__ has a conformal and hence
complex structure through its induced Riemannian metric. In
[16],
Chern proved that an immersed surface in __\( \mathbf{R}^n \)__ is minimal
if and only if the Gauss map is antiholomorphic. This theorem was
proved by
Pinl
for __\( n = 4 \)__ and is the starting point for relating
minimal surfaces with the value distribution theory of
Nevanlinna,
Weyl, and
Ahlfors.
One of the fundamental results of minimal surface
theory is the Bernstein uniqueness theorem, which says that a minimal
graph __\( z = f (x, y) \)__ in __\( \mathbf{R}^3 \)__, defined for all __\( (x, y)
\in \mathbf{R}^2 \)__, must be a plane. Note that the image of the
Gauss map of an entire graph lies in a hemisphere. Bernstein’s theorem
as generalized by
Osserman
says that if the image of the Gauss map of
a complete minimal surface of __\( \mathbf{R}^3 \)__ is not dense, then the minimal
surface is a plane. In
[16],
using a classical theorem of
E. Borel, Chern generalized the Bernstein–Osserman theorem to a density theorem
on the image of the Gauss map of a complete minimal surface in
__\( \mathbf{R}^n \)__, that is not a plane. More refined density theorems
were established in
[18],
a joint paper with Osserman.

Motivated by
Calabi’s
work on minimal 2-spheres in __\( \mathbf{S}^n \)__, Chern
developed in
[20]
a general formalism for osculating spaces for
submanifolds. He proved that given a minimal surface in a space form
there is an integer __\( m \)__ such that the osculating spaces of order __\( m \)__ are
parallel along the surface, and gave a complete system of local
invariants, with their relations. As a consequence, he proved an
analogue of Calabi’s theorem: if a minimal sphere of constant Gaussian
curvature __\( K \)__ in a space form of constant sectional curvature __\( c \)__ is not
totally geodesic, then
__\[ K = \frac{2c}{m(m + 1)} .\]__

##### (2) Tight and taut immersions

We first recall a theorem of
Fenchel,
proved in 1929: if __\( \alpha(s) \)__ is a
simple closed curve in __\( \mathbf{R}^3 \)__, parametrized by its arc length, and __\( k(s) \)__
is its curvature function, then
__\[ \int |k(s)|\,ds \geq 2\pi ,\]__
and equality holds if
and only if __\( \alpha \)__ is a convex plane curve.
Fary and
Milnor
proved that if
__\( \alpha \)__ is knotted then this integral must be greater than __\( 4\pi \)__.

In
[14]
and
[15],
Chern and
Lashof
generalized these results to
submanifolds of __\( \mathbf{R}^n \)__. Let __\( M^m \)__ be a compact
m-dimensional manifold,
__\[ f : M \rightarrow \mathbf{R}^n \]__
an immersion, __\( \nu^1 (M) \)__ the unit normal sphere bundle of __\( M \)__, and __\( dv \)__ the
natural volume element of __\( \nu^1 (M ) \)__. Let
__\[ N : \nu^1 (M ) \rightarrow \mathbf{S}^{n-1} \]__
denote the
normal map, i.e., __\( N \)__ maps the unit normal vector __\( v \)__ at __\( x \)__ to the parallel
unit vector at the origin. Let __\( da \)__ denote the volume element of __\( \mathbf{S}^{n-1} \)__.
Then the Lipschitz–Killing curvature __\( G \)__ on __\( \nu^1(M ) \)__ is defined by the
equation
__\[ N^{\ast} (da) = G\, \mathit{dA} ,\]__
i.e., __\( G(v) \)__ is the absolute value of the
determinant of the shape operator __\( A_v \)__ of __\( M \)__ along the unit normal
direction __\( v \)__. The absolute total curvature __\( \tau (M, f ) \)__ of the immersion __\( f \)__
is the normalized volume of the image of __\( N \)__,
__\[
\tau (M, f ) =\frac{1}{c_{n-1}}\,\int_{\nu^1(M)} |\operatorname{det}(A_v)|\,dv,
\]__
where __\( c_{n-1} \)__ is the volume of the unit __\( (n - 1) \)__-sphere. In
[14]
Chern and Lashof generalized Fenchel’s theorem by showing that
__\[ \tau(M, f ) \geq 2 ,\]__
with equality if and only if __\( M \)__ is a convex
hypersurface of an __\( (m + 1) \)__-dimensional affine subspace __\( V \)__. In
[15]
they obtained the sharper result that
__\[
\tau (M, f ) \geq \sum\beta_i (M ),
\]__
where __\( \beta_i (M ) \)__ is the __\( i \)__-th Betti number of __\( M \)__.

An immersion __\( f : M \rightarrow \mathbf{R}^n \)__ is called tight if
__\( \tau (M, f ) \)__ is equal to the infimum, __\( \tau (M ) \)__, of the absolute
total curvature among all the immersions of M into Euclidean spaces of
arbitrary dimensions. The study of absolute total curvature and tight
immersion has become an important field in submanifold geometry that
has seen many interesting developments in recent years. An important
step in this development is
Kuiper’s
reformulation of tightness in
terms of critical point theory. He showed that for a given compact
manifold __\( M \)__, __\( \tau (M ) \)__ is the Morse number __\( \gamma \)__ of __\( M \)__, i.e.,
the minimum number of critical points a nondegenerate Morse function
must have. Moreover, an immersion of __\( M \)__ is tight if and only if every
nondegenerate height function has exactly __\( \tau (M ) = \gamma \)__
critical points. Another development is the concept of taut immersion
introduced by
Banchoff
and
Carter–West. An immersion of __\( M \)__ into
__\( \mathbf{R}^n \)__ is called taut if every nondegenerate Euclidean
distance function from a fixed point in __\( \mathbf{R}^n \)__ to the
submanifold has exactly __\( \gamma \)__ critical points. Taut implies tight,
and moreover a taut immersion is an embedding. Tautness is invariant
under conformal transformations, hence using stereographic
projection we may assume taut submanifolds lie in the sphere.
Pinkall proved that the tube __\( M_\epsilon \)__ of radius __\( \epsilon \)__ around a
submanifold __\( M \)__ in __\( \mathbf{R}^n \)__ is a taut hypersurface if and
only if __\( M \)__ is a taut submanifold. In particular, this gives two
facts: one is that the parallel hypersurface of a taut hypersurface in
__\( \mathbf{S}^n \)__ is again taut, another is that to understand taut
submanifolds it suffices to understand taut hypersurfaces. Since the
Lie sphere group (the group of contact transformations carrying
spheres to spheres) is generated by conformal transformations and
parallel translations, tautness is invariant under the Lie sphere
group. Note also that the __\( \epsilon \)__-tube __\( M \)__ of a submanifold __\( M \)__ in
__\( S^n \)__ is an immersed Legendre submanifold of the contact manifold of
the unit tangent bundle of __\( S^n \)__. Thus tautness really should be
defined for Legendre submanifolds of the contact manifold of the unit
tangent sphere bundle of __\( \mathbf{S}^n \)__. Chern and
Cecil
make this concept
precise in
[26]
and lay some of the basic differential geometric
groundwork for Lie sphere geometry. There are many interesting
examples of tight and taut submanifolds and many interesting theorems
concerning them. But some of the most basic questions are still
unanswered; for example there are no good necessary and sufficient
conditions known for a compact manifolds to be immersed in Euclidean
space as a tight or taut submanifold, and a complete set of local
invariants for Lie sphere geometry is yet to be found.

#### The generalized Gauss–Bonnet theorem

Geometers tend to make a sharp distinction between “local” and
“global” questions, and it is common not only to regard global
problems as somehow more important, but even to consider local theory
“old-fashioned” and unworthy of serious effort. Chern however has
always maintained that research on these seemingly polar aspects of
geometry must of necessity go hand-in-hand; he felt that one could
not hope to attack the global theory of a geometric structure until
one understood its local theory (i.e., the equivalence problem), and
moreover, once one *had* discovered the local invariants of a theory,
one was well on the way towards finding its global invariants as well!
We shall next explain how Chern came to this contrary attitude, for it
is an interesting and revealing story, involving the most exciting and
important events of his research career: his discovery of an
“intrinsic proof” of the Generalized Gauss–Bonnet Theorem and, flowing
out of that, his solution of the characteristic class problem for
complex vector bundles by his striking and elegant construction of
what are now called “Chern classes” from his favorite raw material,
the curvature forms of a connection. The Gauss–Bonnet Theorem for a
closed, two-dimensional Riemannian manifold __\( M \)__ was surely one of the
high points of classical geometry, and it was generally recognized
that generalizing it to higher dimensional Riemannian manifolds was a
central problem of global differential geometry. The theorem states
that the most basic topological invariant of __\( M \)__, its Euler
characteristic __\( \chi (M ) \)__, can be expressed as __\( 1/2\pi \)__ times the integral over
M of its most basic geometric invariant, the Gaussian curvature
function __\( K \)__. Although there were many published proofs of this, Chern
reproved it for himself by a new method that was very natural from a
moving frames perspective. Moreover, unlike the published proofs,
Chern’s had the potential to generalize to higher dimensions.

To explain Chern’s method, we start by applying the standard moving
frames approach to n-dimensional oriented Riemannian manifolds __\( M \)__,
then specialize to __\( n = 2 \)__. The orientation together with the
Riemannian structure give an __\( \mathbf{SO}(n) \)__ structure for __\( M \)__.
Since the Lie algebra __\( L(\mathbf{SO}(n)) \)__ is just the skew-adjoint
__\( n{\times}n \)__ matrices, in the principal __\( \mathbf{SO}(n) \)__ bundle __\( F
(M ) \)__ of oriented orthonormal frames of __\( M \)__, in addition to the __\( n \)__
canonical 1-forms __\( (\omega_{i}) \)__, we will have the connection 1-forms
for the Levi-Civita connection, a skew-adjoint __\( n {\times} n \)__ matrix of
1-forms __\( \omega_{ij} \)__, characterized uniquely by the zero torsion
condition,
__\[ d\omega_i =\sum_j \omega_{ij} \wedge \omega_j .\]__
The
components __\( R_{ijkl} \)__ of the Riemann curvature tensor in the frame __\( \omega_i \)__
are determined from the curvature forms
__\( \Omega_{ij} \)__ by
__\[ \Omega_{ij} = \frac{1}{2}\sum_{kl}
R_{ijkl} \omega_k \wedge \omega_l \]__
(plus the condition of being skew-symmetric in __\( (i, j) \)__
and in __\( (k, l) \)__).

When __\( n = 2 \)__, the Lie algebra __\( L(\mathbf{SO}(n)) \)__ is
1-dimensional; __\( \omega_{11} = \omega_{22} = 0 \)__ and __\( \omega_{21} =
-\omega_{12} \)__, so there is only one independent __\( \omega_{ij} \)__, namely
__\( \omega_{12} \)__, and so only one curvature equation,
__\[ d\omega_{12} = -\Omega_{12} =
-R_{1212} \,\omega_1 \wedge \omega_2 .\]__
Now it is easily seen that __\( R_{1212} \)__ is a
constant on every fiber __\( \Pi^{-1} (x) \)__, and its value is in fact the Gaussian
curvature __\( K(x) \)__. We can identify the area 2-form, __\( \mathit{dA} \)__, on __\( M \)__ with
__\( -\theta_1 \wedge \theta_2 \)__, where __\( (\theta_1 , \theta_2 ) \)__ is any oriented
orthonormal frame, so that
__\[ \Pi^{\ast}(\mathit{dA}) = -\omega_1 \wedge \omega_2 .\]__
Thus we
can rewrite the above curvature equation as a formula for the
pull-back of the Gauss–Bonnet integrand, __\( K \mathit{dA} \)__, to __\( F (M ) \)__:
__\begin{equation}
\label{ast}
\Pi^\ast (K \mathit{dA}) = d\omega_{12}.
\tag{*}
\end{equation}__
In
[25]
Chern remarks that, along with zero torsion equations, the
formula
__\( (\text{*}) \)__ contains

“… all the information on local Riemannian geometry in two dimensions [and] gives global consequences as well. A little meditation convinces one that

\( (\text{*}) \)must be the formal basis of the Gauss–Bonnet formula, and this is indeed the case. It turns out that the proof of the n-dimensional Gauss–Bonnet formula can be based on this idea….”

Chern noticed a remarkable property of
__\( (\text{*}) \)__. Since
the Gauss–Bonnet integrand is a 2-form on a 2-dimensional manifold, it
is automatically closed, and hence its pull-back under __\( \Pi^{\ast} \)__ must also be
closed. But (except when __\( M \)__ is a torus) __\( K \mathit{dA} \)__ is never exact, so we do
not expect its pull back to be exact. Nevertheless,
__\( (\text{*}) \)__ says that it
is! This phenomenon of a closed but nonexact form on the base of a
fiber bundle becoming exact when pulled up to the total space is
called transgression. As we shall see, it plays a key rôle in Chern’s
proof.

By elementary topology, in the complement __\( M^{\prime} \)__ of any point __\( p \)__ of a
closed Riemannian manifold __\( M \)__ one can always define a smooth vector
field __\( e_1 \)__ of unit length, and the index of this vector field at __\( p \)__
is __\( \chi(M ) \)__. We will now see how this well-known characterization
of the Euler characteristic together with the transgression formula
__\( (\text{*}) \)__ leads quickly to Chern’s proof of the Gauss–Bonnet theorem for
two-dimensional __\( M \)__. Let __\( e_2 \)__ denote the unit length vector field in
__\( M^{\prime} \)__ making __\( (e_1 , e_2 ) \)__ an oriented frame, and let __\( \theta \)__ denote
the dual coframe field in __\( M^{\prime} \)__. Since __\( \Pi \)__ composed with
__\( \theta \)__ is the identity map of __\( M^{\prime} \)__, we have
__\[ d(\theta^{\ast}(\omega_{12} )) = \theta^\ast (d\omega_{12} ) = K \mathit{dA} \]__
in __\( M^{\prime} \)__, so
__\[ \int_M
K \mathit{dA} = \int_{M^{\prime}} K \mathit{dA} =\int_{M^{\prime}} d(\theta^{\ast} (\omega_{12} )) .\]__
If we
write __\( M_\epsilon \)__ for the complement of the open __\( \epsilon \)__-ball about __\( p \)__,
then
__\[ \int_{M^{\prime}}= \lim_{\epsilon\rightarrow 0}\int_{M_\epsilon} ,\]__
and
by Stokes’ Theorem,
__\[ \int_M K \mathit{dA} = \lim_{\epsilon \rightarrow 0}
\int_{S_{\epsilon}} \theta^\ast (\omega_{12} ) ,\]__
where __\( S_\epsilon = \partial M_\epsilon \)__ is the distance sphere
of radius __\( \epsilon \)__ about __\( p \)__. The proof will be complete if we can identify the
right hand side of the latter equation with __\( 2\pi \)__ times the index of __\( \epsilon_1 \)__
at __\( p \)__.

Choose Riemannian normal coordinates in a neighborhood __\( U \)__ of __\( p \)__ and
let __\( (\hat{e}_1 , \hat{e}_2 ) \)__ denote the local frame field in __\( U \)__
defined by orthonormalizing the corresponding coordinate basis
vectors, and __\( \hat{\theta} \)__ the dual coframe field. If __\( \alpha(x) \)__
denotes the angle between __\( e_1(x) \)__ and __\( \hat{e}_1 (x) \)__, then we recall
that the standard expression for the index or winding number of __\( e_1 \)__
with respect to __\( p \)__ is
__\[ \frac{1}{2\pi}\int_C d\alpha \]__
where __\( C \)__ is a small simple closed curve
surrounding __\( p \)__; so we will be done if we can show that the right hand
side above is equal to
__\[ \int_{S_\epsilon} d\alpha .\]__

Let __\( \rho(\alpha) \in \mathbf{SO}(2) \)__ denote rotation through an
angle __\( \alpha \)__. The gauge transformation __\( g : U \rightarrow \mathbf{SO}(2) \)__
from the coframe __\( \hat{\theta} \)__ to the coframe __\( \theta \)__ is just __\( g(x) =
\rho(\alpha(x)) \)__, so by the transformation law for pull-backs of
connection forms noted above,
__\[ \theta^\ast (\omega_{12} ) = d\alpha +
\hat{\theta}^\ast (\omega_{12} ) .\]__
Thus __\( \int_{S_\epsilon} \theta^\ast
(\omega_{12} ) \)__ can be written as the sum of two terms. The first is
the desired __\( \int_{S_\epsilon} d\alpha \)__, and the second term,
__\[ \int_{S_\epsilon} \hat{\theta}^\ast (\omega_{12} ) \]__
clearly tends to zero
with __\( \epsilon \)__ since the integrand is continuous at __\( p \)__, while the length of __\( S_\epsilon \)__
tends to zero.

We now return to the case of a general n-dimensional oriented
Riemannian manifold __\( M \)__ and develop some machinery we will need to
explain the remarkable results that grew out of this approach to
the two-dimensional Gauss–Bonnet Theorem.

A basic problem is how
to construct differential forms on __\( M \)__ canonically from the metric.
Up in the coframe bundle, __\( F (M ) \)__, there is an easy way to
construct differential forms naturally from the metric — simply take
“polynomials” in the curvature forms __\( \Omega_{ij} \)__. Certain forms __\( \Lambda \)__
constructed this way will “define” a form __\( \lambda \)__ on __\( M \)__ by the relation
__\[ \Lambda = \Pi^\ast \lambda ,\]__
and these are the forms we are after.

To make this precise we consider the ring __\( \mathcal{R} \)__ of
polynomials with real (or complex) coefficients in __\( n(n - 1)/2 \)__
variables __\( \{X_{ij}\} \)__, __\( 1 \leq i < j \leq n \)__. We use matrix
notation; __\( X \)__ denotes the __\( n {\times} n \)__ matrix __\( X_{ij} \)__ of elements
of __\( \mathcal{R} \)__, where __\( X_{ji} = -X_{ij} \)__ for __\( i < j \)__, and __\( X_{ii}
= 0 \)__. For __\( g \in \mathbf{SO}(n) \)__,
__\[ \operatorname{ad}(g)X = gXg^{-1} \]__
is the matrix
__\[ \sum_{k,l} g_{ik} X_{kl} g_{jl} \]__
of
elements of __\( \mathcal{R} \)__. If for __\( g \)__ in __\( \mathbf{SO}(n) \)__ and
__\( P \)__ in __\( \mathcal{R} \)__ we define __\( \operatorname{ad}(g)P \)__ in
__\( \mathcal{R} \)__ by
__\[ \bigl(\operatorname{ad}(g)P \bigr)(X) = P
\bigl(\operatorname{ad}(g)X\bigr) ,\]__
this defines an “adjoint” action of
__\( \mathbf{SO}(n) \)__ on __\( \mathcal{R} \)__ (by ring automorphisms). The
subring of “ad-invariant” elements of __\( \mathcal{R} \)__ is denoted by
__\( \mathcal{R}^{\operatorname{ad}} \)__. For future reference we note that
we can also regard __\( X \)__ as representing the general __\( n {\times} n \)__
skew-symmetric matrix, i.e., the general element of the Lie
algebra __\( L(\mathbf{SO}(n)) \)__, and __\( \mathcal{R} \)__ is just the
ring of polynomial functions on __\( L(\mathbf{SO}(n)) \)__.

The curvature 2-forms __\( \Omega_{ij} \)__, being of even degree, commute
with each other under exterior multiplication, so we can
substitute them in elements __\( P \)__ of __\( \mathcal{R} \)__; if __\( P (X) \)__ is
homogeneous of degree __\( d \)__ in the __\( X_{ij} \)__, then __\( P (\Omega) \)__ will be a
differential __\( 2d \)__-form on __\( F (M ) \)__.

Now let __\( \theta \)__ be a local orthonormal coframe field in an open
set __\( U \)__ of __\( M \)__, i.e., a local section __\( \theta : U \rightarrow F (M
) \)__, and let __\( \Psi = \theta^\ast (\Omega ) \)__ denote the matrix of
pulled back curvature forms in __\( U \)__. Since __\( \theta^\ast \)__ is a
Grassmann algebra homomorphism, for any __\( P \)__ in __\( \mathcal{R} \)__,
__\[ \theta^\ast \bigl(P (\Omega)\bigr) = P (\Psi) .\]__
In particular for any __\( x \)__
in __\( U \)__ we have
__\[ \theta^\ast \bigl(P (\Omega )\bigr)_x = P (\Psi_x ) .\]__
If
__\( \hat{\Psi} \)__ is the matrix of curvature forms in __\( U \)__ corresponding
to some other local coframe field, __\( \hat{\theta} \)__ in __\( U \)__, and __\( g
: U \rightarrow \mathbf{SO}(n) \)__ is the change of gauge mapping
__\( \theta \)__ to __\( \hat{\theta} \)__, then as noted
__\[ \hat{\Psi}_x = \operatorname{ad}(g(x))\Psi ,\]__
so we find
__\[ P (\hat{\Psi}_x ) =
\bigl(\operatorname{ad}(g(x))P\bigr)(\Psi_x ) .\]__
Thus in general the pulled back
form __\( P (\Psi) \)__ depends on the choice of __\( \theta \)__ and is only defined
locally, in __\( U \)__. However *if (and only if ) *P* is in the
subring *__\( \mathcal{R}^{\operatorname{ad}} \)__* of
*__\( \operatorname{ad} \)__*-invariant polynomials, the form *__\( P (\Psi) \)__* is a
globally well-defined form on *__\( M \)__*, independent of the choice of local
frame fields *__\( \theta \)__* used to pull back the locally defined curvature
matrices *__\( \Psi \)__. In this case it is clear that
__\[ \Pi^\ast \bigl(P (\Psi)\bigr) = P (\Omega ) ,\]__
a relation that uniquely determines __\( P (\Psi) \)__.

There are many ways one might attempt to generalize the
Gauss–Bonnet Theorem for surfaces, but perhaps the most obvious and
natural is to associate with every compact, oriented, n-dimensional
Riemannian manifold without boundary, __\( M \)__, an __\( n \)__-form __\( \lambda \)__ on
__\( M \)__ that is canonically defined from the metric, and has the property
that
__\[ \lambda = c_n \chi(M ) ,\]__
where __\( c_n \)__ is some universal constant.
If __\( n \)__ is odd then Poincaré duality implies that __\( \chi(M ) = 0 \)__ when
__\( M \)__ is without boundary, and since we will only consider the closed
case here, we will assume __\( n = 2k \)__. (On the other hand, for
odd-dimensional manifolds *with* boundary, the Gauss–Bonnet
Theorem is interesting and decidedly nontrivial!). From the above
discussion it is clear that we should define __\( \lambda = P
(\Psi) \)__, where __\( P \)__ is an ad-invariant polynomial, homogeneous of
degree __\( k \)__ in the __\( X_{ij} \)__. In fact there is an obvious candidate for
__\( P \)__ — the classical Pfaffian, __\( \operatorname{Pf} \)__, uniquely
determined (up to sign) by the condition that
__\[ \operatorname{Pf}(X)^2 = \det(X) \]__
(cf. [e2], p. 309).

A Generalized Gauss–Bonnet Theorem had already been proved in
two papers, one by
Allendoerfer
and the other by
Fenchel.
Both proofs
were “extrinsic” — they assumed __\( M \)__ could be isometrically
embedded in some Euclidean space. (A paper of Allendoerfer and Weil
implied that the existence of local isometric embeddings was enough,
thereby settling the case of analytic metrics). These earlier proofs
wrote the Generalized Gauss–Bonnet integrand as the volume element
times a scalar that was a complicated polynomial in the components of
the Riemann tensor. In
[9]
Chern for the first time wrote the
integrand as the Pfaffian of the curvature forms and then provided a
simple and elegant *intrinsic* proof of the theorem along the
lines of the above proof for surfaces.

Let __\( \mathbf{S}(M ) \)__ denote the bundle of unit vectors of
the tangent bundle to __\( M \)__, and __\( \pi : \mathbf{S}(M ) \rightarrow
M \)__ the natural projection. Given a coframe __\( \theta \)__ in __\( F (M ) \)__ let
__\( e_1 (\theta) \)__ denote the first element of the frame dual to __\( \theta \)__.
Then __\( e_1 : F (M ) \rightarrow \mathbf{S}(M ) \)__ is a fiber bundle
and clearly __\( \Pi : F (M ) \rightarrow M \)__ factors as
__\[ \Pi = \pi \circ e_1 .\]__
Let __\( \lambda \)__ be the __\( n \)__-form __\( \operatorname{Pf}(\Psi) \)__ on __\( M \)__, and
__\( \Lambda = p^\ast (\lambda) \)__ its pull-back to __\( \mathbf{S}(M ) \)__. In
[9]
Chern first proves a transgression lemma for __\( \Lambda \)__, i.e., he
explicitly finds an __\( (n - 1) \)__-form __\( \Theta \)__ on __\( \mathbf{S}(M ) \)__
satisfying __\( d\Theta = \Lambda \)__. As in two dimensions let __\( M^{\prime} \)__ be the
complement of some point __\( p \)__ in __\( M \)__ and construct a smooth
cross-section __\( \xi \)__ of __\( \mathbf{S}(M ) \)__ over __\( M \)__. Then __\( \pi\circ
\xi \)__ is the identity map of __\( M^{\prime} \)__, so just as in the two dimensional
argument we find
__\[ d(\xi^\ast (\theta)) = \lambda ,\]__
and
__\[ \int_M = \lim_{\epsilon \rightarrow\theta}\int_{S_\epsilon} \xi^\ast (\Theta) .\]__
Finally, the construction of __\( \Theta \)__ is so explicit that Chern is
able to evaluate the right hand side by an argument similar to the one
in the surface case, and he finds that it is indeed a universal
constant times the Euler characteristic of __\( M \)__.

Mathematicians in general value proofs of new facts much more highly than elegant new proofs of old results. It is worth commenting why [9] is an exception to this rule. The earlier proofs of the Generalized Gauss–Bonnet Theorem were virtually a dead end while, as we shall see below, Chern’s intrinsic proof was a key that opened the door to the secrets of characteristic classes.

#### Characteristic classes

The coframe bundle, __\( F (M ) \)__, that
keeps reappearing in our story, is an important example of a
mathematical structure known as a *principal *__\( G \)__*-bundle*. These
were first defined and their study begun only in the late 1930s, but
their importance was quickly recognized by topologists and geometers,
and the theory underwent intensive development during the 1940s. By
the end of that decade the beautiful classification theory had been
worked out, and with it the related theory of “characteristic
classes”, a concept whose importance for the mathematics of the
latter half of the twentieth century it would be difficult to
exaggerate. (As we will see below, in the language we have been using,
the classification problem is the equivalence problem for principal
bundles, and characteristic classes are invariants for this
equivalence problem).

In order to explain Chern’s role in these important developments we will first review some of the basic mathematical background of the theory.

We will consider only the case of a Lie group __\( G \)__. Since the theory is
essentially the same for a Lie group and one of its maximal compact
subgroups, we will also assume that __\( G \)__ is compact. A “space” will
mean a paracompact topological space, and a __\( G \)__-space will mean a
space, __\( P \)__, together with a continuous right action of __\( G \)__ on __\( P \)__.
We will write __\( R_g \)__ for the homeomorphism __\( p \mapsto pg \)__. The
__\( G \)__-space __\( P \)__ is called a *principal *__\( G \)__*-bundle* if the action
is free, i.e., if for all __\( p \)__ in __\( P \)__, __\( R_g (p) \neq p \)__ unless __\( g \)__ is the
identity element __\( e \)__ of __\( G \)__. More specifically, __\( P \)__ is called a
principal __\( G \)__-bundle over a space __\( X \)__ if we are given some fixed
homeomorphism of __\( X \)__ with the orbit space __\( P/G \)__, or equivalently if
there is given a “projection map”
__\[ \Pi : P \rightarrow X \]__
such that
the __\( G \)__ orbits of __\( P \)__ are exactly the “fibers” __\( \Pi^{-1} (x) \)__ of
the map __\( \Pi \)__. __\( P \)__ is called the total space of the bundle, and we
often denote the bundle by the same symbol as the total space. A map
__\( \sigma : X \rightarrow P \)__ that is a left inverse to __\( \Pi \)__ is called
a section. Two __\( G \)__-bundles over __\( X \)__, __\( \Pi_i : P_i \rightarrow X \)__, __\( i
= 1, 2 \)__ are considered “equivalent” if there is a __\( G \)__-equivariant
homeomorphism
__\[ \varphi : P_1 \rightarrow P_2
\quad\text{such that}\quad
\Pi_1 =
\Pi_2 \circ \varphi .\]__
The principal __\( G \)__-bundle over __\( X \)__ defined by
__\( P = X \times G \)__ with __\( R_g (x, \gamma) = (x, \gamma g) \)__ and __\( \Pi(x,
\gamma) = x \)__ is called the product bundle, and any bundle equivalent
to the product bundle is called a trivial bundle. Clearly __\( x \mapsto
(x, e) \)__ is a section of the product bundle, so any trivial bundle has
a section. Conversely, if __\( \Pi : P \rightarrow X \)__ has a section
__\( \sigma \)__, then __\( \varphi(x, g) = R_g (\sigma(x)) \)__ is an equivalence of
the product bundle with __\( P \)__, i.e., *a principal *__\( G \)__*-bundle is
trivial if and only if it admits a section*. We will denote the set of
equivalence classes [P ] of principal __\( G \)__-bundles __\( P \)__ over __\( X \)__ by
__\( \operatorname{Bndl}_G (X) \)__.

Given a principal __\( G \)__-bundle __\( \Pi : P \rightarrow X \)__ and a continuous
map __\( f : Y \rightarrow X \)__, we can define a bundle __\( f^\ast (P ) \)__ over
__\( Y \)__, called the bundle induced from __\( P \)__ by the map __\( f \)__. Its total
space is
__\[ \bigl\{(p, y) \in P \times Y \bigm| \Pi(p) = f (y)\bigr\} ,\]__
with the
projection __\( (p, y) \mapsto y \)__ and the __\( G \)__-action
__\[ R_g (p, y) = \bigl(R_g (p), y\bigr) .\]__
It is easy to see that __\( f^\ast \)__ maps equivalent bundles to
equivalent bundles, so it induces a map (also denoted by __\( f^\ast \)__)
from __\( \operatorname{Bndl}_G (X) \)__ to __\( \operatorname{Bndl}_G (Y ) \)__. If
__\( \Pi : P \rightarrow X \)__ is a principal __\( G \)__-bundle then __\( \Pi^\ast (P ) \)__
is a principal __\( G \)__-bundle over the total space __\( P \)__, called the
“square” of the original bundle. In fact this bundle is always
trivial, since it admits the “diagonal” section
__\( p \mapsto (p, p) \)__. As we will see below, this simple
observation is the secret behind
transgression!

The first nontrivial fact in the theory is the so-called “covering
homotopy theorem”; it says that the induced map
__\[ f^\ast :
\operatorname{Bndl}_G (X) \rightarrow \operatorname{Bndl}_G (Y ) \]__
depends only on the homotopy class __\( [f ] \)__ of __\( f \)__. We can paraphrase
this by saying that __\( \operatorname{Bndl}_G (\,\cdot\,) \)__ is a contravariant
functor from the category of spaces and homotopy classes of maps to
the category of sets. Now a cohomology theory is also such a functor,
and a *characteristic class* for __\( G \)__-bundles can be defined as
simply a natural transformation from __\( \operatorname{Bndl}_G (\,\cdot\,) \)__ to
some cohomology theory __\( H^\ast (\,\cdot\,) \)__. Of course this fancy language
isn’t essential and was only invented about the same time as bundle
theory. It just says that a characteristic class __\( c \)__ is a function
that assigns to each principal __\( G \)__-bundle __\( P \)__ over any space __\( X \)__ an
element __\( c(P ) \)__ in __\( H^\ast (X) \)__, with the “naturality” property that
__\[ c\bigl(f^\ast (P )\bigr) = f^\ast \bigl(c(P )\bigr) ,\]__
for any continuous __\( f : Y
\rightarrow X \)__. We fix some cohomology theory __\( H^\ast (\,\cdot\,) \)__ and denote
by __\( \operatorname{Char}(G) \)__ the set of all characteristic classes for
__\( G \)__-bundles. Since __\( H^\ast (X) \)__ has the structure of a ring with unit,
so does __\( \operatorname{Char}(G) \)__, and the characteristic class problem
for __\( G \)__ is the problem of explicitly identifying this ring. Note that
a trivial bundle is induced from a map to a space with one point, so
all its characteristic classes (except the unit class) must be zero.
More generally, equality of all characteristic classes of a bundle is
a necessary (and in some circumstances sufficient) test for their
equivalence, and this is one of the important uses of characteristic
classes.

The remarkable and beautiful classification theorem for principal
__\( G \)__-bundles “solves” the classification problem at least in the
sense of reducing it to a standard problem of homotopy theory. Given
spaces __\( X \)__ and __\( Z \)__ let __\( [X, Z] \)__ denote the set of homotopy classes of
maps of __\( X \)__ into __\( Z \)__. Note that __\( [\,\cdot\, , Z] \)__ is a contravariant
functor, much like __\( \operatorname{Bndl}_G \)__ — any map __\( f : Y
\rightarrow X \)__ induces a pull-back map
__\[ f^\ast : [h] \mapsto [h \circ f ] \]__
of __\( [X, Z] \)__ to __\( [Y, Z] \)__. Moreover if __\( \Pi : P \rightarrow Z \)__ is
any principle __\( G \)__-bundle then we have a map
__\[ [h] \mapsto [h^\ast (P)] \]__
of __\( [X, Z] \)__ to __\( \operatorname{Bndl}_G (X) \)__ that is “natural”
(i.e., it commutes with all “pull-back” maps __\( f^\ast \)__). We call __\( P \)__
a universal principal __\( G \)__-bundle if the latter map is bijective. The
heart of the classification theorem is the fact that universal
__\( G \)__-bundles do exist. In fact it can be shown that a principal
__\( G \)__-bundle is universal provided its total space is contractible, and
there are even a number of methods for explicitly constructing such
bundles.

We will denote by __\( \mathcal{U}_G \)__ some choice of universal principal
__\( G \)__-bundle. Its base space will be denoted by __\( \mathcal{B}_G \)__ and is
called the classifying space for __\( G \)__. (Although __\( \mathcal{B}_G \)__ is not
unique, its homotopy type is). If __\( \Pi : P \rightarrow X \)__ is any
principal __\( G \)__-bundle then, by definition of universal, there is a
*unique* homotopy class __\( [h] \)__ of maps of __\( X \)__ to __\( \mathcal{B}_G \)__
such that __\( P \)__ is equivalent to __\( h^\ast (\mathcal{U}_G ) \)__. Any
representative __\( h \)__ is called a classifying map for __\( P \)__. Clearly if __\( f :
Y \rightarrow X \)__ then __\( h \circ f \)__ is a classifying map for __\( f^\ast (P
) \)__. Also, the classifying map for __\( \mathcal{U}_G \)__ is just the identity
map of __\( \mathcal{B}_G \)__.

It is now easy to give a solution of sorts to the characteristic class
problem for __\( G \)__; namely __\( \operatorname{Char}(G) \)__ is canonically
isomorphic to __\( H^\ast (\mathcal{B}_G ) \)__. In fact each __\( c \in H^\ast
(\mathcal{B}_G ) \)__ defines a characteristic class (also denoted by __\( c \)__)
by the formula
__\[ c(P ) = f^\ast (c) ,\]__
where __\( f \)__ is a classifying map
for __\( P \)__, and the inverse map is just __\( c \mapsto c(\mathcal{U}_G ) \)__.

This is a distillation of ideas developed between 1935 and 1950 by
Chern, Ehresmann, Hopf,
Feldbau, Pontryagin,
Steenrod,
Stiefel, and
Whitney.
While elegant in its simplicity, the above version is still
too abstract and general to be of use in finding
__\( \operatorname{Char}(G) \)__ for a specific group __\( G \)__. It is also of
little use in calculating the characteristic classes of bundles that
come up in geometric problems, for it is not often an easy matter to
find a classifying map from geometric data. We shall discuss how Chern
put flesh on these bones by finding concrete models for classifying
spaces and, more importantly, by showing how to calculate explicitly
de Rham theory representatives of many characteristic classes from the
curvature forms of connections.

Let __\( \mathbf{V}(n, N + n) \)__ denote the *Stiefel manifold* of
__\( n \)__-frames in __\( \mathbf{R}^{N +n} \)__, consisting of all orthonormal
sequences
__\[ e = (e_1 , \dots, e_n) \]__
of vectors in __\( \mathbf{R}^{N
+n} \)__. There is an obvious free action of __\( \mathbf{O}(n) \)__ on
__\( \mathbf{V}(n, N + n) \)__, and the orbit of __\( e \)__ consists of all
__\( n \)__-frames spanning the same __\( n \)__-dimensional linear subspace that __\( e \)__
does. Thus we have an __\( \mathbf{O}(n) \)__ principal bundle
__\[ \Pi : \mathbf{V}(n, N +n) \rightarrow \mathbf{Gr}(n, N +n) ,\]__
where
__\( \mathbf{Gr}(n, N +n) \)__ is the Grassmannian of all __\( n \)__-dimensional
linear subspaces of __\( \mathbf{R}^{N +n} \)__. In the early 1940s it
was known from results of Steenrod and Whitney that this bundle is
“universal for compact k-dimensional polyhedra”, provided __\( N \geq k
+ 1 \)__. This means that for any compact polyhedral space __\( X \)__, with
__\( \dim(X) \leq k \)__, every principal __\( \mathbf{O}(n) \)__ bundle over __\( X \)__
is of the form
__\[ h^\ast \bigl(\mathbf{V}(n, N +n)\bigr) \]__
for a unique __\( [h] \)__ in __\( [X,
\mathbf{Gr}(n, N + n)] \)__. In
[12]
Chern and
Y. F. Sun
generalized
these results to show that this bundle is also universal for compact
__\( k \)__-dimensional ANRs. (If one wants universal bundles in the strict
sense described above, one need only form the obvious inductive limit,
__\[ \Pi : \mathbf{V}(n, \infty) \rightarrow \mathbf{Gr}(n,
\infty) ,\]__
by letting __\( N \)__ tend to infinity. But for the finite
dimensional problems of geometry it is preferable to stick with these
finite dimensional models). By replacing the real numbers respectively
by the complex numbers and the quaternions, Chern and Sun proved
analogous results for the other classical groups __\( \mathbf{U}(n) \)__
and __\( \mathbf{Sp}(n) \)__. They went on to note that if __\( G \)__ is any
compact Lie group, then by taking a faithful representation of __\( G \)__ in
some __\( \mathbf{O}(n) \)__, __\( \mathbf{V}(n, N +n) \)__ becomes a
principal __\( G \)__ bundle by restriction, and the corresponding orbit space
__\[ \mathbf{V}(n, N + n)/G \]__
becomes a classifying space
__\( \mathcal{B}_G \)__ for compact ANRs of dimension __\( \leq k \)__.

The Grassmannians make good models for classifying spaces, for they
are well-studied explicit objects whose cohomology can be
investigated using both algebraic and geometric techniques. From such
computations Chern knew that there was an n-dimensional “Euler
class” __\( e \)__ in __\( \operatorname{Char}(\mathbf{SO}(n)) \)__. If
__\( M \)__ is a smooth, compact, oriented __\( n \)__-dimensional manifold then __\( e(F
(M )) \in H^n (M ) \)__ when evaluated on the fundamental class of __\( M \)__ is
just __\( \chi(M ) \)__. One can thus interpret the Generalized Gauss–Bonnet
Theorem as saying that __\( \lambda = \operatorname{Pf}(\Psi) \)__
represents __\( e(F (M )) \)__ in de Rham cohomology. This inspired Chern to
look for a general technique for representing characteristic classes
by de Rham classes. This was in 1944–1945, while Chern was in
Princeton, and he discussed this problem frequently with his friend
André Weil who encouraged him in this search.

It might seem natural to start by trying to represent
__\( \mathbf{SO}(n) \)__ characteristic classes by closed
differential forms, but Chern made what was to be a crucial
observation: the cohomology of the real Grassmannians is complicated.
In particular it contains a lot of __\( \mathbf{Z}_2 \)__ torsion, and
this part of the cohomology is invisible to de Rham theory. On the
other hand Chern knew that Ehresmann, in his thesis, had calculated
the homology of complex Grassmannians and showed there was no torsion.
In fact Ehresmann showed that certain explicit algebraic cycles (the
“Schubert cells”) form a free basis for the homology over
__\( \mathbf{Z} \)__. It follows from de Rham’s Theorem that all the
cohomology classes for __\( \mathcal{B}_{\mathbf{U}(n)} \)__ can be
represented by closed differential forms. These forms, when pulled
back by the classifying map of a principal __\( \mathbf{U}(n) \)__-bundle,
will then represent the characteristic classes of the bundle in
de Rham cohomology. While this is fine in theory, it still depends on
knowing a classifying map, while what is needed in practice is a
method to calculate these characteristic forms from geometric data. We
now explain Chern’s beautiful algorithm for doing this.

Let __\( \Pi : P \rightarrow M \)__ be a smooth principle
__\( \mathbf{U}(n) \)__-bundle over a smooth manifold __\( M \)__. Recall that a
connection for __\( P \)__ can be regarded as a 1-form __\( \omega \)__ on __\( P \)__ with values in
the Lie algebra of __\( \mathbf{U}(n) \)__, __\( L(\mathbf{U}(n)) \)__, which
consists of all __\( n {\times} n \)__ skew-Hermitian complex matrices.
Equivalently we can regard __\( \omega \)__ as an __\( n {\times} n \)__ matrix of
complex-valued 1-forms __\( \omega_{ij} \)__ on __\( P \)__ satisfying __\( \omega_{ji} =
-\bar{\omega}_{ij} \)__,
and similarly for the associated curvature 2-forms __\( \Omega_{ij} \)__.

We will denote by __\( \mathcal{R} \)__ the ring of complex-valued polynomial
functions on the vector space __\( L(\mathbf{U}(n)) \)__. Using the usual
basis for the __\( L(\mathbf{U}(n)) \)__, we can identify __\( \mathcal{R} \)__
with complex polynomials in the __\( 2n(n - 1) \)__ variables __\( X_{ij} \)__, __\( Y_{ij} \)__,
__\( 1 \leq i < j \leq n \)__ and the __\( n \)__ variables __\( Y_{ii} \)__, __\( 1 \leq i \leq
n \)__. __\( Z \)__ will denote the __\( n {\times} n \)__ matrix of elements in
__\( \mathcal{R} \)__ defined by
__\begin{align*}
& Z_{ij} = X_{ij} + \sqrt{-1}\,Y_{ij},\\
& Z_{ji} = -X_{ij} + \sqrt{-1}\,Y_{ij}\quad\text{and}\\
& Z_{ii} = \sqrt{-1}\,Y_{ii}
\end{align*}__
for __\( 1
\leq i < j \leq n \)__. We can also regard __\( Z \)__ as representing the general
element of __\( L(\mathbf{U}(n)) \)__, and we will write __\( Q(Z) \)__ rather
than __\( Q(X_{ij} , Y_{ij}) \)__ to denote elements of __\( \mathcal{R} \)__. The
adjoint action of the group __\( \mathbf{U}(n) \)__ on its Lie algebra
__\( L(\mathbf{U}(n)) \)__ is now given by
__\[ \operatorname{ad}(g)(Z) =
gZg^{-1} ,\]__
just as in the __\( \mathbf{SO}(n) \)__ case above,
and as in that case we define the adjoint action of
__\( \mathbf{U}(n) \)__ on __\( \mathcal{R} \)__ by
__\[ \bigl(\operatorname{ad}(g)Q\bigr)(Z) =
Q\bigl(\operatorname{ad}(g)Z\bigr) .\]__
As before we denote by
__\( \mathcal{R}^{\operatorname{ad}} \)__ the subring of __\( \mathcal{R} \)__
consisting of ad invariant polynomials. Once again we can substitute
the curvature forms __\( \Omega_{ij} \)__ for the __\( Z_{ij} \)__ in an element
__\( Q(Z) \)__ in __\( \mathcal{R} \)__, and obtain a differential form __\( Q(\Omega) \)__ on
__\( P \)__; if __\( Q \)__ is homogeneous of degree __\( d \)__ in its variables then
__\( Q(\Omega) \)__ is a __\( 2d \)__-form. The same argument as in the
__\( \mathbf{SO}(n) \)__ case shows that if __\( Q \in
\mathcal{R}^{\operatorname{ad}} \)__ then __\( Q(\Omega) \)__ is the pull-back of
a uniquely determined form __\( Q(\Psi) \)__ on __\( M \)__. Using the Bianchi
identity, Chern showed that
__\[ \mathit{dQ}(\Psi) = 0 \]__
(cf. [e2], p. 297),
so
__\( Q(\Psi) \)__ represents an element __\( [Q(\Psi)] \)__ in __\( H^\ast (M ) \)__, the
complex de Rham cohomology ring of __\( M \)__. If we use a different
connection __\( \omega \)__ on __\( P \)__ with curvature matrix __\( \Omega \)__ then we get
a
different closed form __\( Q(\Psi ) \)__ on __\( M \)__ with
__\[ \Pi^\ast \bigl(Q(\Psi^{\prime})\bigr) = Q(\Omega^{\prime}) .\]__
What is the
relation between __\( Q(\Psi^{\prime} ) \)__ and __\( Q(\Psi) \)__? Weil provided Chern with the
necessary lemma: they differ by an exact form, so that __\( [Q(\Psi)] \)__ is a
well-defined element of __\( H^\ast (M ) \)__, independent of the connection. We
will denote it by __\( \hat{Q}(P ) \)__. (Weil’s lemma can be derived as a corollary
of the fact that __\( Q(\Psi) \)__ is closed. For the easy but clever proof see
[e2],
p. 297).

If __\( h : M \rightarrow M \)__ is a smooth map, then a connection on __\( P \)__
“pulls-back” naturally to one on the __\( \mathbf{U}(n) \)__-bundle __\( h^\ast (P ) \)__ over __\( M \)__.
The curvature forms likewise are pull-backs, from which it is
immediate that
__\[ Q\bigl(h^\ast (P )\bigr) = h^\ast \bigl(Q(P )\bigr) .\]__
In other words, __\( Q \mapsto \hat{Q} \)__ is a
map from __\( \mathcal{R}^{\operatorname{ad}} \)__ into __\( \operatorname{Char}(\mathbf{U}(n)) \)__. It is clearly a ring homomorphism, and in
recognition of Weil’s lemma Chern called it the Weil homomorphism, but
it is more commonly referred to as the Chern–Weil homomorphism.

For __\( \mathbf{U}(n) \)__ the ring __\( \mathcal{R}^{\operatorname{ad}} \)__ of
ad-invariant polynomials on its Lie algebra has an elegant and
explicit description that follows easily from the diagonalizability of
skew-Hermitian operators and the classic classification of symmetric
polynomials. Extend the adjoint action of __\( \mathbf{U}(n) \)__ to the
polynomial ring __\( \mathcal{R}[t] \)__ by letting it act trivially on the
new indeterminate __\( t \)__. The characteristic polynomial
__\[ \operatorname{det}(Z + tI) = \sum_{k=0}^n \sigma_k (Z)\,t^{n-k} \]__
is
clearly ad-invariant, and hence its coefficients __\( \sigma_k (Z) \)__ belong
to __\( \mathcal{R}^{\operatorname{Ad}} \)__. Substituting a particular matrix
for __\( Z \)__ in __\( \sigma_k (Z) \)__ gives the __\( k \)__-th elementary symmetric
function of its eigenvalues; in particular __\( \sigma_1 (Z) =
\operatorname{trace}(Z) \)__ and __\( \sigma_n (Z) = \det(Z) \)__. Now if
__\[ P (t_1, \dots, t_n ) \in \mathbf{C}[t_1 , \dots, t_n ] \]__
then of course
__\[ P \bigr(\sigma_1 [Z],\dots, \sigma_n [Z]\bigl) \]__
is also in
__\( \mathcal{R}^{\operatorname{ad}} \)__. In fact,
__\[ \mathcal{R}^{\operatorname{ad}} = C[\sigma_1, \dots,\sigma_n ] ,\]__
i.e.,
__\[ P (t_1 , \dots, t_n ) \mapsto P \bigl(\sigma_1 [Z], \dots, \sigma_n
[Z]\bigr) \]__
*is a ring isomorphism*. From this fact, together with
Ehresmann’s explicit description of the homology of complex
Grassmannians, Chern was easily able to verify that
*the Chern–Weil homomorphism is in fact an isomorphism
of *__\( \mathcal{R}^{\operatorname{ad}} \)__* with *__\( \operatorname{Char}(\mathbf{U}(n)) \)__.
For technical reasons it
is convenient to renormalize the polynomials __\( \sigma_k (Z) \)__, defining
__\[ \gamma_k (Z) = \sigma_k \Bigl( \frac{1}{2\pi i} Z\Bigr) .\]__
Then we get a
__\( \mathbf{U}(n) \)__-characteristic class __\( c_k = \hat{\gamma}_k \)__ of
dimension __\( 2k \)__, called the __\( k \)__-th Chern class, and these __\( n \)__ classes
__\( c_1 , \dots \)__, __\( c_n \)__ are polynomial generators for the characteristic
ring __\( \operatorname{Char}(\mathbf{U}(n)) \)__; that is each
__\( \mathbf{U}(n) \)__-characteristic class __\( c \)__ can be written uniquely as
a polynomial in the Chern classes.

If __\( F (Z) \)__ is a formal power series, __\( F = \sum^\infty_0 F_r \)__, where
__\( F_r \)__ is a homogeneous polynomial of degree __\( r \)__, then for finite
dimensional spaces, __\( \hat{F}_r \)__ will vanish for large __\( r \)__ so
__\[ \hat{F}
= \sum^\infty_0\hat{F}_r \]__
will be a well-defined characteristic class.
Many important classes were defined in this way by
Hirzebruch,
and Chern used the power series
__\[ E(Z) = \operatorname{trace}\Bigl(\exp\Bigl(
\frac{1}{2\pi i} Z\Bigr)\Bigr) \]__
to define the *Chern character*,
__\[ \mathbf{ch} = \hat{E} .\]__
It plays a vital role in the
Atiyah–Singer Index Theorem.

Chern also developed a generalization of the Chern–Weil homomorphism
for an arbitrary compact Lie group __\( G \)__. The adjoint action of __\( G \)__ on
its Lie algebra __\( L(G) \)__ induces one on the ring __\( \mathcal{R} \)__ of
complex-valued polynomial functions on __\( L(G) \)__, so we have a subring
__\( \mathcal{R}^{\operatorname{ad}} \)__ of adjoint invariant polynomials.
Substituting curvature forms of __\( G \)__-connections on __\( G \)__-principal
bundles into such invariant polynomials __\( Q \)__, we get as above a
Chern–Weil homomorphism __\( Q \mapsto Q \)__ of
__\( \mathcal{R}^{\operatorname{ad}} \)__ to the characteristic ring
__\( \operatorname{Char}(G) \)__ (defined with respect to complex de Rham
cohomology) and this is again an isomorphism. Of course, for general
__\( G \)__ the homology of the classifying space __\( \mathcal{B}_G \)__ will have
torsion, so there will be other characteristic classes beyond those
picked up by de Rham theory. Moreover the explicit description of the
ring of adjoint invariant polynomials is in general fairly
complicated.

Chern left the subject of characteristic classes for nearly twenty
years, but then returned to it in 1974 in a now famous joint paper
with
J. Simons
[21].
This paper is a detailed and elegant study of
the phenomenon of transgression in principal bundles. Let __\( M \)__ be an
__\( n \)__-dimensional smooth manifold, __\( \Pi : P \rightarrow M \)__ a smooth
principal __\( G \)__-bundle over __\( M \)__, __\( \omega \)__ a __\( G \)__-connection in __\( P \)__, and
__\( \Omega \)__ the matrix of curvature 2-forms. Given an adjoint invariant
polynomial __\( Q \)__ on __\( L(G) \)__, homogeneous of degree
__\( \ell \)__, we have a globally defined closed __\( 2\ell \)__-form __\( Q(\Psi) \)__ on
__\( M \)__ that represents the characteristic class __\( \hat{Q}(P ) \in
H^{2\ell} (M ) \)__, and that is characterized by
__\[ \Pi^{\ast} \bigl(Q(\Psi)\bigr)
= Q(\Omega) .\]__
Chern and Simons first point out the simple reason why
__\( Q(\Omega) \)__ *must* be an exact form on __\( P \)__. Indeed, by the naturality of
characteristic classes under pull-back, __\( Q(\Omega) \)__ represents
__\( \hat{Q}(\Pi^{\ast} (P )) \)__. But as we saw earlier, __\( \Pi^{\ast} (P
) \)__, the “square” of the bundle __\( P \)__, is a principal __\( G \)__-bundle over
__\( P \)__ with a global cross-section, hence it is trivial and all of its
characteristic classes must vanish. In particular
__\[ \hat{Q}\bigl(\Pi^{\ast} (P )\bigr) = 0 ,\]__
i.e., __\( Q(\Omega) \)__ is exact.

They next write down an explicit formula in terms of __\( Q \)__, __\( \omega \)__,
and __\( \Omega \)__ for a __\( (2 \ell- 1) \)__-form __\( \mathit{TQ}(\omega) \)__ on __\( P \)__,
and show that
__\[ d\mathit{TQ}(\omega) = Q(\Omega) .\]__
__\( \mathit{TQ}(\omega) \)__ is natural under pull-back of a bundle and its connection.
Now suppose __\( 2 > n \)__. Then __\( Q(\Psi) = 0 \)__, so of course __\( Q(\Omega) = 0 \)__,
i.e., in this case __\( \mathit{TQ}(\omega) \)__ is closed, and so defines
an element __\( [\mathit{TQ}(\omega)] \)__ of __\( H^{2\ell -1} (P ) \)__. If __\( 2\ell
> n + 1 \)__ Chern and Simons show this cohomology class is independent of
the choice of connection __\( \omega \)__, and so defines a “secondary
characteristic class”. However if __\( 2 = n + 1 \)__ then they show that
__\( [\mathit{TQ}(\omega)] \)__ does depend on the choice of connection
__\( \omega \)__.

They now consider the case __\( G =\mathbf{GL}(n, \mathbf{R}) \)__
and consider the adjoint invariant __\( n \)__ polynomials __\( Q_k \)__ defined by
__\[ \det(X + tI) =\sum^n_{i=0} Q_i (X)\,t^{n-i} .\]__
Taking __\( Q = Q_{2k-1} \)__
they again show __\( Q(\Omega) = 0 \)__ provided __\( \omega \)__ restricts to an
__\( \mathbf{O}(n) \)__ connection on an __\( \mathbf{O}(n) \)__-subbundle of
__\( P \)__, so of course in this case too we have a cohomology class
__\( [\mathit{TQ}(\omega)] \)__. They specialize to the case that __\( P \)__ is the bundle of
bases for the tangent bundle of __\( M \)__ and __\( \omega \)__ is the Levi-Civita
connection of a Riemannian structure. Then __\( [\mathit{TQ}(\omega)] \)__ is defined, but
depends in general on the choice of Riemannian metric. Now they prove
a remarkable and beautiful fact — __\( [\mathit{TQ}(\omega)] \)__ is invariant
under conformal changes of the Riemannian metric! Such conformal
invariants have recently been adopted by physicists in formulating
so-called conformal quantum field theories.

Chern also returned to the consideration of characteristic classes and
transgression in another joint paper, this one with
R. Bott
[19].
Here
they consider holomorphic bundles over complex analytic manifolds,
where there is a refined exterior calculus, using the __\( \partial \)__ and
__\( \bar{\partial} \)__ operators, and they prove a transgression formula for
the top Chern form of a Hermitian structure with respect to the
operator __\( i\partial \bar{\partial} \)__. This work has applications both to
complex geometry (especially the study of the zeros of holomorphic
sections), and to algebraic number theory. In recent years it has
played an important role in papers by
J. M. Bismut,
H. Gillet,
and
C. Soulé.

#### “Retirement”

For most mathematicians, retirement is a one-time event followed by a period of declining mathematical activity. But as with so much else, Chern’s attitude towards retirement is highly nonstandard. Both authors remember well attending a series of enjoyable so-called retirement parties for Chern, as he retired first from UC Berkeley, then several years later as Director of MSRI, etc. But in each case, instead of retiring, Chern merely replaced one demanding job with another.

Finally, in 1992, Dr. Hu Guo-Ding took over as director of the Nankai Institute of Mathematics and Chern declared himself truly retired. In fact though, he travels back to Nankai one or more times each year and continues to play an active role in the life of the Institute. The Institute now has an excellent library, has become increasingly active in international exchanges, and has many well-trained younger members. In 1995, the occasion of the tenth anniversary of the Nankai Institute was celebrated with a highly successful international conference, attended by many well-known physicists and mathematicians.

Chern also continues to be very active in mathematical research, and
when asked why he doesn’t slow down and take it a little easier, his
stock “excuse” is that he does not know how to do something else. He
says he tries to work in areas that he feels have a future, avoiding
the current fashions. His recent interests have been Lie sphere
geometry, several complex variables, and particularly Finsler
geometry. Chern’s interest in the latter subject has a long history.
Already in 1948 he solved the equivalence problem for the subject in
“Local Equivalence and Euclidean connections in Finsler spaces”
(reprinted in
[28]).
Chern feels that the time is now ripe to recast
all the beautiful global results of Riemannian geometry of the past
several decades in the Finsler context, and he points out that
thinking of Riemannian geometry as a special case of Finsler geometry
was already advocated by David Hilbert in his twenty-third problem at
the turn of the last century. Chern himself has recently taken some
steps in that direction, in “On Finsler geometry” (*C. R. Acad. Sci.
Paris*, t. 314, Série I, p. 757–761, 1992), and with David Bao, “On
a notable connection in Finsler geometry” (*Houston Journal
of Math.*, v. 19, no. 1, 1993). He has also recently spelled out the
general program in a paper that is as yet unpublished, “Riemannian
geometry as a special case of Finsler geometry”.