#### by Hung-Hsi Wu

Shiing-shen Chern, one of the giants in the history of geometry, passed away in Tianjin, China, on December 3, 2004, at the age of ninety-three. He had spent the last five years of his life in Tianjin, but his career was established mainly in the U.S. from 1949 to 1999.

Chern’s work revitalized and reshaped differential geometry and transcendental algebraic geometry. In the decades before 1944 when he embarked on the writing of his historic papers on Chern classes and the geometry of fiber bundles, the field of differential geometry had gone through a period of stagnation. His papers marked the re-entry of differential geometry into the mathematical mainstream, and his tenure at Berkeley (1960–1979) helped make the latter a premier center of geometry in the world. In his lifetime, he had the pleasure of seeing Chern classes become part of the basic vocabulary in contemporary mathematics and theoretical physics. There are also Chern–Weil homomorphism, refined Chern classes for Hermitian bundles, Chern–Moser invariants, and Chern–Simons invariants. Chern was also a mathematical statesman. One does not often see great mathematical insight and great political leadership converge on the same person, but Chern was that rare exception. He was the main liaison between the American and Chinese mathematics communities in the years immediately following the re-opening of China in 1978. Moreover, in the four decades from 1946 to 1984, he founded or co-founded three mathematics institutes; the one in the U.S., the Mathematical Sciences Research Institute, is thriving in Berkeley.

He was born on October 26, 1911, in Jiaxing, Zhejiang Province,
China, sixteen days after
the revolution that overthrew the Manchurian Dynasty and ushered in
modern China. Typically for that era in China,
his schooling was haphazard. He had *one day* of elementary education, four
years of middle-high school, and at age fifteen skipped two grades to
enter Nankai University. Under the tutelage of a great
teacher in Nankai, he began to learn some substantive mathematics.
To the end of his life, he retained a fond memory of his Nankai years,
and this fact was to play a role in his eventual decision to return to China
in 1999. His next four years
(1930–34) were spent as a graduate student at Qing Hua University
in Beijing (then *Peiping*), and he published three papers on the
subject of projective differential geometry. While he sensed that his future
lay in the general area of geometry, he reminisced in his later years
that he was at a loss “how to climb this beautiful mountain.” In 1932,
Wilhelm Blaschke
of Hamburg, Germany, came to visit Beijing
and gave some lectures on web geometry. Although those
lectures were quite elementary,
they opened his eyes to the fact that the era of projective
differential geometry had come and gone, and something vaguely called “global
differential geometry” was beckoning on the horizon. When he won a
scholarship in 1934 to study abroad, he defied the conventional wisdom of
going to the U.S. and chose to
attend the University of Hamburg instead. This is the
first of three major decisions he made in the ten-year period 1934–1943
that shaped the rest of his life. As we shall see, the decision in each case
was by no means easy or obvious, but it only appeared to be so
with hindsight. Throughout his life, he never seemed to lose this
uncanny ability to make the right decision at the right time.

Soon after his arrival at Hamburg, he solved one of Blaschke’s problems in
web geometry and was awarded a doctorate in 1936. However, the most important
discovery he made during his two years in Hamburg was the work of
Élie Cartan.
The discovery was due to not only
the fact that Blaschke was one of the few at the time
who understood and recognized the importance of Cartan’s geometric work, but
also the happy coincidence that
E. Kähler
had just
published what we now call the Cartan–Kähler theory on exterior differential
systems, and was giving a seminar on this theory at Hamburg.
When Chern was given a postdoctoral fellowship
in 1936 to pursue further study
in Europe, he sought Blaschke’s advice. The latter presented
him with two choices: either stay in Hamburg to learn algebraic number
theory from
Emil Artin,
or go to Paris to learn geometry from Élie Cartan.
At the time, Artin was a major star; he was also a
phenomenal teacher, as Chern knew very well firsthand. But Chern made his
second major decision
by choosing Élie Cartan and Paris. His one-year stay in Paris (1936–37) was,
in his own words, “unforgettable.” He got to know the master’s
work directly from the master himself, and Cartan’s influence on his
scientific outlook can be seen on almost every page of his four-volume
*Selected Papers* (1978–1989).

Even before he returned to China in 1937, Chern had been appointed professor of mathematics at Qing Hua University, his former graduate school. Unfortunately, the Sino-Japanese War broke out in Northeast China when he was still in Paris, and Qing Hua University was moved to Kunming in southwestern China as part of the Southwest Associated University. It was to be ten more years before he could set his eyes again on the Qing Hua campus in Beijing. During 1937–43, he taught and studied in isolation in Kunming under harsh war conditions. It must be said that sometimes a little isolation is not a bad thing for people engaging in creative work. For Chern, those years broadened and deepened his understanding of Cartan’s work. He wrote near the end of his life that as a result of his isolation, he got to read over 70% of Cartan’s papers which total 4,750 pages. Another good thing that came out of those years was his marriage to Shih-ning Cheng in 1939, although a few months later, his pregnant bride had to leave him to return to Shanghai for reasons of personal safety. Their son Paul was born the following year but did not get to meet his father until he was six years old.

In all those years, he kept up his research and his papers
appeared in international journals,
including two in the *Annals of Mathematics* in 1942.
Of the latter, the one on integral geometry
[1]
was reviewed
in the *Mathematical Reviews* by
André Weil
who gave it high praise, and the other on
isotropic surfaces
[2]
was refereed by none other than
Hermann Weyl,
who
made this fact known to Chern himself when they finally met
1n 1943. Weyl read every
line of the manuscript of
[2],
made suggestions for improvement, and recommended it with enthusiasm.
But Chern was not satisfied with just being a known quantity to
the mathematical elite because he wanted
to find his own mathematical
voice. When invitation to visit the Institute for Advanced
Study (IAS) at Princeton came from
O. Veblen
and Weyl
in 1943, he seized the opportunity and
accepted in spite of the hardship of wartime travel.
At considerable personal risk, he spent seven days to fly by military
aircrafts from Kunming to Miami via India, Africa,
and South America. He reached Princeton in August by train.
The visit to IAS was his third major decision of the preceding decade,
and perhaps the most important of all.

His sojourn at the IAS from August 1943 to December of 1945 changed
the course of differential geometry and transcendental algebraic
geometry; it changed his whole life as well.
Soon after his arrival at Princeton, he made a discovery that not only
solved one of the major problems of the
day — to
find an *intrinsic* proof of the __\( n \)__-dimensional Gauss–Bonnet
theorem — but also
enabled him to define Chern classes on principal bundles with structure
group __\( U(n) \)__, the unitary group.
In the crudest terms, the discovery in question is that, on a
Riemannian or Hermitian manifold, the curvature form of the metric
can be used to generate topological invariants in a canonical way:
certain polynomials in the curvature form are closed differential forms, and are
therefore cohomology classes of the manifold via de Rham’s theorem.
The best entry to this circle of ideas is still Chern’s first paper on the subject, the
remarkable six-page paper in the
*Annals of Mathematics* on the intrinsic
proof of the Gauss–Bonnet theorem
[3].
We now turn to a brief description of this paper
(cf. [e10]).

Let __\( M \)__ be a compact oriented Riemannian manifold of dimension __\( 2n \)__. Let
a local *frame field* __\( e_1, \dots,e_{2n} \)__ be chosen (i.e., the __\( e_i \)__
are locally defined vector fields and are orthonormal with respect to the
metric). Let __\( \{ \omega^i \} \)__ be the *dual co-frame field* of
__\( \{ e_i \} \)__ (i.e., the __\( \omega^i \)__ are 1-forms defined in the same
neighborhood as the __\( e_i \)__ and __\( \omega^i(e_j) = \delta^i_j \)__ for __\( i, j =
1, \dots, 2n \)__). Furthermore, let __\( \Omega^i_j \)__ be the *curvature form*
with respect to __\( \{ e_i \} \)__. Note that if __\( \tilde{e}_1, \dots, \tilde{e}_{2n} \)__
is another frame field and __\( h \)__ is the orthogonal transition matrix
between the frame fields,
__\begin{equation}
\tilde{e}_i = \sum_k e_k h^k_i ,
\end{equation}__
then the curvature form
__\( \tilde{\Omega}^i_j \)__ with respect to __\( \{ \tilde{e}_i \} \)__ satisfies
__\[ \tilde{\Omega}^i_j = \sum_{\ell, k} (h^{-1})^i_{\ell} \Omega^{\ell}_{k}
h^k_j \]__
But __\( (h^{-1})^i_{\ell} = h^{\ell}_i \)__, therefore we have
__\begin{equation}
\label{1}
\tilde{\Omega}^i_j = \sum_{\ell, k} h^{\ell}_i \Omega^{\ell}_{k} h^k_j
\end{equation}__
Define the following __\( 2n \)__-form __\( \Omega \)__ by
__\begin{equation}
\label{2}
\Omega= \frac{1}{2^{2n} \pi^n n!} \sum \varepsilon_{i_1 \dots i_{2n}}
\Omega^{i_1}_{i_2} \dots \Omega^{i_{2n-1}}_{i_{2n}}
\end{equation}__
where __\( \varepsilon_{i_1 \dots i_{2n}} \)__ is __\( +1 \)__ or __\( -1 \)__, depending on whether
__\( i_1 \dots i_{2n} \)__ is an even or odd permutation of __\( 1, \dots , 2n \)__,
and is otherwise equal to 0. This __\( \Omega \)__ is, a priori, dependent on the
choice of __\( \{ e_i \} \)__ and is therefore defined only in the neighborhood where
__\( \{ e_i \} \)__ is defined. Equation __\eqref{1}__, however, implies that if __\( \{ e_i \} \)__
is replaced by __\( \{ \tilde{e}_i \} \)__, so that __\( \Omega^i_j \)__ is replaced
by the curvature form __\( \tilde{\Omega}^i_j \)__ corresponding to
__\( \{ \tilde{e}_i \} \)__, then
__\[ \sum \varepsilon_{i_1 \dots i_{2n}}
\Omega^{i_1}_{i_2} \dots \Omega^{i_{2n-1}}_{i_{2n}} =
\sum \varepsilon_{i_1 \dots i_{2n}}
\tilde{\Omega}^{i_1}_{i_2} \dots \tilde{\Omega}^{i_{2n-1}}_{i_{2n}}.\]__
Therefore the form __\( \Omega \)__ is independent of the the choice of the frame
field __\( \{ e_i \} \)__ and is a globally defined __\( 2n \)__-form on __\( M \)__. The
Gauss–Bonnet
theorem, first proved in
complete generality by
Allendoerfer–Weil
[e1],
in 1943,
states that
__\begin{equation}
\label{3}
\int_M \Omega = \chi(M)
\end{equation}__
where __\( \chi (M) \)__ denotes the Euler characteristic of __\( M \)__. We shall refer to
__\( \Omega \)__ as the *Gauss–Bonnet integrand*.

The problem with the Allendoerfer–Weil proof is that it is conceptually
complex: as the phrase “Riemannian polyhedra” in the
title of
[e1]
suggests, it begins by triangulating __\( M \)__
into a simplicial complex with small simplices which are (essentially)
isometrically imbeddable into Euclidean space, then integrates the
Gauss–Bonnet integrand over each simplex (here earlier results on the
Gauss–Bonnet theorem by
Fenchel
and Allendoerfer for submanifolds in
Euclidean space are employed), and then add up the results for the individual
simplices carefully to make sure that the boundary terms cancel
and the Euler characteristic emerges. One does
not know at the end of the proof *why* the theorem is true. Weil conveyed
his own misgivings about the proof
to Chern upon the latter’s arrival at Princeton, and suggested
to him that there must be a proof that is *intrinsic* in the sense
of not having to appeal to imbedding into Euclidean space. Chern’s proof in
[3]
achieved exactly this goal, and we proceed to sketch its main ideas.

Consider the *frame bundle* __\( F(M) \)__ of __\( M \)__,
which is the fibre bundle of orthonormal bases,1
__\begin{multline}
F(M)=\bigl\{ (x,f_1, \dots, f_{2n}): x\in M, \text{ and } f_1, \dots, f_{2n} \text{ are}\\
\text{an orthonormal basis in the tangent space of } M \text{ at } x \bigr\} .
\end{multline}__
We have the projection map __\( \pi: F(M) \to M \)__. Since
__\( \Omega \)__ is a form on __\( M \)__, the pull-back __\( \pi^{\ast} \Omega \)__
is a form on __\( F(M) \)__. The major step of Chern’s proof of the
Gauss–Bonnet theorem is that there exists a __\( (2n-1) \)__-form __\( \Pi \)__ on __\( F(M) \)__
so that
__\begin{equation}
\label{4}
\pi^{\ast} \Omega = d\Pi
\end{equation}__
Thus the Gauss–Bonnet integrand, when pulled back to __\( F(M) \)__, becomes
*exact*! This was a totally unexpected result, and was one that
underscored for the first time the intrinsic importance of fibre bundles
*in differential geometry*. Now sometimes a surprising fact
can turn out to be
rather trivial because it may only depend on a simple trick, but
__\eqref{4}__
is quite
the opposite. Let __\( \theta^i_j \)__ be the connection form of the Levi-Civita
connection on __\( F(M) \)__; __\( \theta^i_j \)__ (__\( i,j=1, \dots, 2n \)__)
is a 1-form with value in the
skew-symmetric matrices __\( \mathfrak{so}(2n) \)__, the Lie algebra of the
special orthogonal
group __\( \mathrm{SO}(2n) \)__. The curvature form __\( \Theta^i_j \)__ on __\( F(M) \)__, a 2-form also
taking value in __\( \mathfrak{so}(2n) \)__, is given by
__\[ \Theta^i_j = d\theta^i_j + \sum_k \theta^i_k \wedge \theta^k_j .\]__
This __\( \Theta^i_j \)__ is related to the preceding __\( \Omega^i_j \)__ as follows: If
__\( \{ e_i \} \)__ is the local frame field as before and __\( \Omega^i_j \)__ is the
curvature form with respect to __\( \{ e_i \} \)__, let __\( e \)__ be the local cross-section
of __\( \pi: F(M) \to M \)__ defined by
__\[ e(x) = (x, e_1(x), \dots e_{2n}(x)) .\]__
Then __\( e^{\ast} (\Theta^i_j) = \Omega^i_j \)__ for all __\( i \)__, __\( j \)__.

The last fact about __\( \Theta^i_j \)__ and __\( \Omega^i_j \)__ has the following
consequence. Consider the __\( 2n \)__-form __\( \Theta \)__ defined on __\( F(M) \)__ by
__\begin{equation}
\label{5}
\Theta = \frac{1}{2^{2n} \pi^n n!} \sum \varepsilon_{i_1 \dots i_{2n}}
\Theta^{i_1}_{i_2} \dots \Theta^{i_{2n-1}}_{i_{2n}}
\end{equation}__
where __\( \varepsilon_{i_1 \dots i_{2n}} \)__ has the same meaning as before. It follows
from __\eqref{2}__ that __\( e^{\ast} \Theta = \Omega \)__, so that __\( (e\circ \pi)^{\ast} \Theta
= \pi^{\ast} \Omega \)__. A simple reasoning2
shows that __\( (e\circ \pi)^{\ast} \Theta =
\Theta \)__. Combining these two relations, we get:
__\begin{equation}
\label{6}
\pi^{\ast} \Omega = \Theta
\end{equation}__
Thus to prove
__\eqref{4}__,
it suffices to prove
__\begin{equation}
\label{7}
\Theta= d \Pi
\end{equation}__
for some __\( (2n-1) \)__-form __\( \Pi \)__ on __\( F(M) \)__.

Chern’s proof of
__\eqref{7}__
requires the introduction, on __\( F(M) \)__, of the following
__\( (2n-1) \)__-forms __\( \Phi_0 \)__, __\( \Phi_1 \)__, …, __\( \Phi_{n-1} \)__, and the __\( 2n \)__-forms
__\( \Psi_0 \)__, __\( \Psi_1 \)__, …, __\( \Psi_{n-1} \)__: for each __\( k=0, \dots n-1 \)__,
__\[\Phi_k = \sum \varepsilon_{i_1, \dots, i_{2n-1}} \Theta^{i_1}_{i_2} \wedge
\dots \wedge \Theta^{i_{2k-1}}_{i_{2k}} \wedge \theta^{i_{2k+1}}_{i_{2n}}
\wedge \dots \wedge \theta^{i_{2n-1}}_{i_{2n}}\]__
and
__\[\Psi_k = (2k+1) \sum \varepsilon_{i_1, \dots, i_{2n-1}}
\Theta^{i_1}_{i_2} \wedge
\dots \wedge \Theta^{i_{2k-1}}_{i_{2k}} \wedge \Theta^{i_{2k+1}}_{i_{2n}}
\wedge \theta^{i_{2k+2}}_{i_{2n}}
\wedge \dots \wedge \theta^{i_{2n-1}}_{i_{2n}}\]__
where each sum is over all permutations __\( i_1, \dots, i_{2n-1} \)__ of 1, …,
__\( 2n-1 \)__, and __\( \varepsilon_{i_1, \dots, i_{2n-1}} \)__ is equal to __\( +1 \)__ or __\( -1 \)__,
depending on whether the permutation is even or odd. Note that from __\eqref{5}__,
we have
__\begin{equation}
\label{8}
\Psi_{n-1} = (2^{2n} \pi^n n!) \Theta
\end{equation}__
Using the Bianchi identity and the definition of __\( \Theta^i_j \)__ in terms of
the connection form __\( \theta^i_j \)__, one obtains the following recurrence
relation:
__\begin{equation}
\label{9}
d\Phi_k = - \Psi_{k-1} + \frac{2n-2k-1}{2(k+1)} \Psi_k
\end{equation}__
where __\( k=0, \dots, n-1 \)__ and __\( \Psi_{-1} \equiv 0 \)__ by definition. The
sought-after __\( (2n-1) \)__-form __\( \Pi \)__ on __\( F(M) \)__ is now defined to be
__\begin{equation}
\label{10}
\Pi = \frac{1}{\pi^n} \sum^{n-1}_{k=0}
\frac{1}{1\cdot 3\cdot 5 \dots (2n-2k-1) \cdot 2^{n+k} k!} \Phi_k
\end{equation}__
Using __\eqref{9}__, and then __\eqref{8}__, we finally get
__\[d\Pi = \frac{1}{2^{2n} \pi^n n!} \Psi_{n-1} = \Theta,\]__
which is exactly __\eqref{7}__.

This proof of __\eqref{7}__ is the envy and despair of all who work in differential
geometry. Chern did this computation mainly in his head,3
and all through his life,
he seemed to be able to conjure at will the same
magical quality in his computations.

For the concluding step in the Chern proof of __\eqref{3}__, we have to bring in the
*sphere bundle*
__\[ S(M)= \bigl\{(x,f):
f \text{ is a unit vector in the tangent space of } M \text{ at }x \bigr\} .\]__
__\( F(M) \)__ is a fibre bundle over
__\( S(M) \)__ and we have a natural projection __\( \pi_1 : F(M) \to S(M) \)__.
Briefly, the forms __\( \Phi_k \)__ and __\( \Psi_k \)__
actually *descend* to __\( S(M) \)__, in the sense that they are the pull-backs
of forms in __\( S(M) \)__ by __\( \pi^{\ast}_1 \)__. The same is therefore true of
__\( \Pi \)__ and __\( \Theta \)__, so that we may regard __\eqref{7}__ as a relation between forms
on __\( S(M) \)__. Now given any point __\( x_0 \)__ in __\( M \)__,
the Hopf theorem on vector fields says there is a unit vector
field __\( v \)__ defined in __\( M\setminus \{x_0\} \)__ so that its isolated singularity
at __\( x_0 \)__ has index equal to __\( \chi (M) \)__, the Euler characteristic of __\( M \)__.
Regarding __\( v \)__ as a cross-section of the bundle __\( S(M)\to M \)__ over
__\( M\setminus \{x_0\} \)__, it is elementary to see that, on __\( M\setminus \{x_0\} \)__,
__\[ \Omega = d (v^{\ast} \Pi) \]__
Moreover, and this is a critical observation due to Chern, the restriction
of __\( \Pi \)__ (as a form on __\( S(M) \)__) to the fibre __\( S_{x_0} \)__ of __\( S(M) \)__ over
__\( x_0 \)__ is exactly
__\[\Pi \mid_{S_{x_0}} = \frac{(n-1)!}{2 \pi^n} d \sigma\]__
where __\( d\sigma \)__ is the volume form of the unit sphere __\( S_{x_0} \)__ in the tangent
space of __\( M \)__ at __\( x_0 \)__.4
By
a standard argument, expressing __\( M\setminus \{x_0\} \)__
as the limit of __\( M \)__ minus the small ball of radius __\( \varepsilon \)__ around __\( x_0 \)__
as __\( \varepsilon \to 0 \)__ and using Stokes’ theorem, the
integral of __\eqref{3}__ becomes the integral
__\[ \int_{S_{x_0}} v^{\ast} \Pi = \int_{v_{\ast}(S_{x_0})} \Pi = \chi (M)\]__
where we have made use of the classical fact that the volume of the
unit sphere in __\( 2n \)__-dimensional space is __\( 2\pi^n/(n-1)! \)__.
*Now we see exactly why the Gauss–Bonnet theorem is true.*

We may interpret the preceding proof in the following way. The form __\( \Omega \)__,
being a top degree form on __\( M \)__, is automatically closed and therefore
represents a cohomology class by de Rham’s theorem. The Gauss–Bonnet theorem
__\eqref{3}__ says is that this class is the Euler class. Here then is the first example
of a canonical representation of a cohomology class by the curvature form
of a Riemannian metric. Once this is realized, the next step is perfectly
obvious, i.e., how to generalize this construction.
The fact that this was Chern’s thinking can be inferred
not only from his 1946
paper
[5]
which introduces Chern classes,
but more explicitly from what he said
concerning the Gauss–Bonnet theorem in the last sentence of the second
paragraph on p. 85 of
[5]
and also from pp. 114–115, *loc. cit.* Before
commenting further on
[5],
however, let us pause to make a few
historical remarks.

The whole idea of using the curvature form on a principal bundle to generate
characteristic classes is now so standard that it is difficult for us,
sixty years after the fact, to fully appreciate the startling originality
of Chern’s contribution. The fact revealed by
__\eqref{4}__,
to the effect that
in differential geometry,
the associated bundles of a manifold are part and parcel of any attempt
to understand the manifold itself, was unimagined at the time. The use
of the curvature form and de Rham’s theorem to generate cohomology classes
was equally revelatory. Perhaps the words of a contemporary, André Weil,
can more accurately give a sense of Chern’s accomplishment. Weil was among
the first in his generation to recognize the significance of
Élie Cartan’s
work, and was familiar with Cartan’s theory of exterior differential forms
as well as Cartan’s use of fibre bundles.
In fact,
Weil
originally wanted to write the
Allendoerfer–Weil paper
[e1]
using differential forms instead of
tensors (Weil
[e8], p. 554).
Thus he had every advantage
a mathematician could ask for to
decipher the Gauss–Bonnet enigma, but the insight that there would be a vast
conceptual simplification of the Gauss–Bonnet integrand by use of the
sphere bundle (in the form of
__\eqref{4}__),
and that the integrand is a representative
of a cohomology class eluded him. As he noted:

Les espaces fibrés … Leur rôle en géométrie différentielle, et tout particulièrement dans l’oeuvre d’Élie Cartan a été longtemps resté implicite, mais s’était clarifié peu à peu grâce aux travaux d’Ehresmann et surtout à ceux de Chern. La démonstration par Chern de la formule de Gauss–Bonnet et sa découverte des classes caractéristiques des variétés à structure complexe ou quasi-complexe avaient inauguré une nouvelle époque en géométrie différentielle, … [e8], p. 566.

[Chern and I] were both beginning to realize the major role which fibre bundles were playing, still mostly behind the scenes, in all kinds of geometric problems. … I will merely point out what can now be realized in retrospect about Chern’s proof for the Gauss–Bonnet theorem, as compared with the one Allendoerfer and I had given in 1942, following the footsteps of H. Weyl and other writers. The latter proof, resting on the consideration of “tubes,” did depend (although this was not apparent at the time) on the construction of a sphere-bundle, but of a non-intrinsic one, viz., the transversal bundle for a given immersion in Euclidean space; Chern’s proof operated explicitly for the first time with an intrinsic bundle, the bundle of tangent vectors of length 1, thus clarifying the whole subject once and for all. [e7], p. x–xi.

These passages may also shed some light on why Weil’s admiration of Chern never flagged throughout his life.

It was already mentioned that Chern began his quest for defining general
characteristic classes almost as soon as he saw how to prove the Gauss–Bonnet
theorem. To cut a long story short, the result of this work is the
substance of his paper
[5].
Briefly,
let a Hermitian metric be given on an __\( n \)__-dimensional complex
manifold __\( M \)__, and let __\( \Omega^i_j \)__ (__\( i, j =1,\dots n \)__)
be the curvature form of the
Hermitian connection relative to a local unitary frame field __\( \{ e_i \} \)__
(__\( i=1,\dots n \)__) (i.e., each __\( e_i \)__ is a vector field of type __\( (1,0) \)__, and
__\( \{ e_i(x) \} \)__ is an orthonormal basis of the holomorphic tangent space
at __\( x \)__ for each __\( x \)__ with respect to the Hermitian metric). __\( \Omega^i_j \)__
is of type __\( (1,1) \)__.
Consider now the following n differential forms __\( c_k (\Omega) \)__
(__\( k=1, \dots n \)__) of type __\( (k,k) \)__:
__\begin{equation}
\label{11}
c_k (\Omega) = \Bigl(\frac{\sqrt{-1}}{2 \pi}\Bigr)^k
\sum_{i_1 < i_2 < \dots < i_k} \sum_{\sigma} \varepsilon(\sigma)\,
\Omega^{i_1}_{\sigma(i_1)} \wedge \dots \Omega^{i_k}_{\sigma(i_k)}
\end{equation}__
where each __\( \sigma \)__ ranges through all permutations of __\( i_1 \)__, …, __\( i_k \)__,
and the corresponding __\( \varepsilon(\sigma) \)__ is the sign of the permutation,
which is __\( +1 \)__ if __\( \sigma \)__ is even, and __\( -1 \)__ if
__\( \sigma \)__ is odd. These are
the *Chern forms* of the hermitian metric. One argues as in __\eqref{2}__ above
that these __\( c_k (\Omega) \)__ do not depend on the choice of the unitary
frame field __\( \{ e_i \} \)__ so that they are globally defined differential
forms on __\( M \)__.
A computation using the Bianchi identity shows that in fact each __\( c_k (\Omega) \)__
is a closed form. By de Rham’s theorem, each __\( c_k (\Omega) \)__
represents a cohomology class of degree __\( 2k \)__, the __\( k \)__-th *Chern class*
of the manifold __\( M \)__. The unusual looking coefficient in __\eqref{11}__ guarantees
that the Chern classes are integral classes.

Now let
__\begin{multline}
U(M)= \bigl\{(x,u_1, \dots, u_n): x\in M, \text{ and } u_1, \dots, u_n \text{ are }\\
\text{an orthonormal basis of the holomorphic tangent space of } M \text{ at } x \bigr\} .
\end{multline}__
We shall refer to __\( U(M) \)__ as the *bundle of unitary frames*
over __\( M \)__. We have the natural projection __\( \pi: U(M) \to M \)__.
The analogue of __\eqref{4}__ is that each of these forms __\( c_k(\Omega) \)__, when pulled
back to __\( U(M) \)__, becomes an exact form.5
As in __\eqref{10}__, this fact is proved
by an explicit construction:
__\begin{equation}
\label{12}
\pi^{\ast} c_k (\Omega) = d (Tc_k(\Omega))
\end{equation}__
where each __\( Tc_k(\Omega) \)__ is a form explicitly constructed from __\( c_k (\Omega) \)__.
For simplicity, we shall refer to
__\( Tc_k(\Omega) \)__ as the *transgression* of __\( c_k(\Omega) \)__.6
Below, we shall have occasion to refer to the fact
that each __\( Tc_k(\Omega) \)__ can be written down explicitly in terms of
__\( c_k (\Omega) \)__ and the metric.

When the Hermitian metric is Kählerian, Chern identified the
__\( n \)__-th Chern form __\( c_n(\Omega) \)__
with the Gauss–Bonnet integrand of the underlying *Riemannian* manifold of
__\( M \)__
([5], pp. 114–5).
Thus
one sees the direct link between the papers
[3]
and
[5].
(As is well known, the __\( n \)__-th Chern *class* is always the Euler class;
see
[e6].)
Moreover, Chern’s definition of the forms __\( c_k(\Omega) \)__ in
__\eqref{11}__
is based
on the fact that the polynomials corresponding to
the __\( c_k(\Omega) \)__ generate the invariant polynomials of the unitary
group. Thus in Chern’s seminal work, we see the key ingredients of Weil’s
1949 definition of the aptly named *Chern–Weil homomorphism* on
a general fibre bundle with an arbitrary Lie group as structure group
([e8], pp. 422–436).

To round off the picture, it should be pointed out that the analogue of the Chern forms for the orthogonal group was introduced around the same time by Pontryagin [e2], though the details came later [e3].

The topology of the forties was preoccupied with the real category, and Chern’s work on the characteristic classes of complex manifolds appeared at first to be slightly out of step with the times. But the dramatic growth of algebraic geometry, particularly transcendental algebraic geometry, beginning with the fifties made him a prophet. Chern classes are important in algebraic geometry for at least two reasons. One is that the Chern classes of algebraic varieties suggested that they might furnish a firm foundation for the (then) confusing plethora of algebraic-geometric invariants, and Hodge was among the first to push for this point of view [e4]. Chern himself made important contributions in this direction, but F. Hirzebruch’s work in the fifties capped this development and made this vision a reality [e5]. A second and perhaps more important reason is that, many by-now standard arguments in algebraic geometry (e.g., those using the Kodaira vanishing theorem or applications of Yau’s solution of the Calabi Conjecture) are simply not possible without the curvature representations of the Chern classes of a bundle.

Chern’s fame began to spread after 1944, though
slowly, in the American mathematics community,
and he was invited to give a one-hour address in the 1945 summer
meeting of the American Mathematical Society. In reviewing the text of
that address
[4],
Heinz Hopf
wrote in *Mathematical Reviews* that Chern’s
work had ushered in a new era in global differential geometry.
Thereafter, the global study of manifolds became the main direction
of geometric research. At age thirty-four, he had realized his youthful
dream by scaling one of the highest peaks on that “beautiful mountain.”

In April of 1946, Chern returned to China and was immediately entrusted with the creation of a mathematics institute for Academia Sinica in Nanking. That he did, and became its de facto director (the official title was “Deputy Director”). We normally envision a “mathematics institute” to be a gathering of scholars to explore the frontiers of research, but China was not yet ready for that kind of institute for lack of a sufficient number of such Chinese mathematical scholars. Being a realist from beginning to end, Chern turned the institute into the only thing it could have been, namely, China’s first true graduate school in mathematics. He recruited a group of young people and personally took charge of their education by teaching them the fundamentals of modern mathematics. Many of this group subsequently became leaders of the next generation of Chinese mathematicians.

By late 1948, the political situation in China had become so unstable that Veblen and Weyl began to be concerned about Chern’s safety. With the help of R. Oppenheimer, then director of IAS, Chern and his family managed to land safely on U.S. soil on New Year’s Day of 1949. He was to be a member of IAS for the spring semester and, in the fall, take up a faculty position at the University of Chicago where he would stay until 1960. In 1950, he gave a one-hour address at the International Congress of Mathematicians (held in Cambridge, Massachusetts) on the differential geometry of fibre bundles. It was in the decade of the fifties that Chern classes began to force their way into most mathematicians’ consciousness, due in no small part to the spectacular advances in algebraic geometry made by Kodaira, Hirzebruch, and others.

In 1960, Chern accepted the offer to come to the University of California at Berkeley. Upon his arrival, he immediately attracted a group of young geometers, and Berkeley in the sixties and seventies became the de facto geometry center of the world. Although he officially retired in 1979, he remained active in Berkeley’s departmental affairs until the mid-eighties, and made Berkeley his home until 1999. Many honors came his way during the Berkeley years, the principal ones being the election to the National Academy of Sciences in 1961, the U.S. National Medal of Science in 1975, and the Wolf Prize from the Israel government in 1984. Later, he also received the Lobachevsky Prize from the Russian Academy in 2002, and the first Shaw Prize in mathematics in 2004, a few months before his death. In 2002, he was Honorary President of the International Congress of Mathematicians held at Beijing.

Chern’s leadership position in differential geometry was, if anything, enhanced
by his work in his Berkeley years. Two of his major papers in this period
hark back to his early work on characteristic classes. On the latter, he
was wont to point out that his main contribution to
characteristic classes was not so much the introduction of Chern classes as
the discovery of explicit
differential *forms* that represent those *classes*.
To him, it was the forms that give geometers an edge over topologists
in studying many aspects of these classes. With examples like
Yau’s
solution of the Calabi Conjecture in mind, one can hardly disagree
with him. The two pieces of work to be discussed
further justify his point of view.
In his collaboration with
Raoul Bott
[7]
in 1965
on generalized Nevanlinna theory in higher dimensions, they
constructed for the holomorphic category the “correct” version of
transgression
(cf. __\eqref{12}__)
in the top dimension by proving that, in case
of an __\( n \)__-dimensional holomorphic vector bundle __\( \pi: E \to M \)__ over an
__\( n \)__-dimensional complex manifold __\( M \)__, the pull-back of the top Chern
form __\( \pi^{\ast} c_n(\Omega) \)__ to __\( E\setminus 0 \)__ (here 0 stands for the
zero section) is not only exact
(see __\eqref{12}__),
but “doubly exact”:
__\[ \pi^{\ast} c_n(\Omega) = dd^c \rho\]__
for some __\( (n-1,n-1) \)__ form __\( \rho \)__ on __\( E\setminus 0 \)__. The nontriviality
of this assertion comes from the fact that __\( E\setminus 0 \)__ is neither
compact nor assumed to be Kählerian. This property of
the top Chern class is crucial for their generalization of Nevanlinna’s
first main theorem. Along the way, they also made use of this
“doubly exact” phenomenon to
introduce the *refined Chern classes* which have since found their way
into algebraic number theory. Incidentally, this Bott–Chern paper is
also a natural extension of Chern’s ground-breaking work of
the fifties to geometrize Nevanlinna theory by
transplanting it to complex manifolds
[6].
The geometric point of view
towards Nevanlinna theory has proven to be extraordinarily fruitful in
algebraic geometry, and it has repercussions in number theory as well.

The second paper related to Chern’s earlier work on
characteristic classes dates from 1971, when
he and
Jim Simons
introduced the Chern–Simons invariants
[9].
Let __\( M \)__ be an __\( n \)__-dimensional Riemannian manifold. We will be appealing to
the Chern–Weil homomorphism, so let
__\( P(u^i_j) \)__ be an invariant polynomial on the Lie algebra __\( \mathfrak{so}(n) \)__ of
the orthogonal group __\( O(n) \)__. If __\( \Theta^i_j \)__ is the curvature form on the
frame bundle __\( F(M) \)__ as in equation __\eqref{5}__ above, then __\( P(\Theta^i_j) \)__ is a closed
from on __\( F(M) \)__ that is the pull-back of a form on __\( M \)__ (compare
equation __\eqref{6}__) and therefore represents
a cohomology class of __\( M \)__. Moreover, as in equations __\eqref{4}__ and __\eqref{12}__,
__\( P(\Theta^i_j) \)__ is actually
an exact form on __\( F(M) \)__, which we write as
__\begin{equation}
\label{13}
P(\Theta^i_j) = dTP(\Theta^i_j)
\end{equation}__
As mentioned earlier, the form __\( TP(\Theta^i_j) \)__ is obtained from
__\( P(\Theta^i_j) \)__ by an explicit construction.
**\textit**{If the form __\( P(\Theta^i_j) \)__ is equal to 0}, then __\( TP(\Theta^i_j) \)__
becomes a closed form on __\( F(M) \)__, and therefore
defines a cohomology class of __\( F(M) \)__ (rather than __\( M \)__ itself).

So far, the forms __\( P(\Theta^i_j) \)__ and __\( TP(\Theta^i_j) \)__ depend on
the choice of the Riemannian metric on __\( M \)__.
Now suppose we change the metric on __\( M \)__ by a conformal factor, then there is
a natural bundle isomorphism between the two frame bundles. In particular,
the cohomology groups of their total spaces are naturally isomorphic and
will henceforth be identified. With this understood, Chern
and Simons proved that, under such a conformal change of metric,
the *form* __\( P(\Theta^i_j) \)__ *does not change*,
and the *cohomology class* of __\( TP(\Theta^i_j) \)__, as a class on __\( F(M) \)__,
does not change either. This cohomology class
__\( [TP(\Theta^i_j)] \)__ is then a conformal
invariant of the Riemannian metric. They went on to give
applications of this fact to conformal immersions into Euclidean space.

In case __\( M \)__ is a 3-dimensional Riemannian manifold and __\( P_1 \)__ is the first
Pontryagin polynomial, then __\( P_1(\Theta^i_j) \)__, being the pull-back of a
4-form on a 3-dimensional manifold, must be 0. The above considerations
therefore apply, and we have a cohomology class __\( [TP_1(\Theta^i_j)] \)__
on __\( F(M) \)__ which is a conformal invariant of __\( M \)__. But in this case, the
*form* __\( TP_1(\Theta^i_j) \)__ can be simply written down: with
__\( \theta^i_j \)__ as the connection form on __\( F(M) \)__ (see the notational setup
preceding equation __\eqref{5}__),
__\begin{equation}
\label{14}
TP_1(\Theta^i_j) = \frac{1}{4 \pi^2}
\bigl\{\theta^1_2 \wedge \theta^1_3
\wedge \theta^2_3 + \theta^1_2 \wedge \Theta^1_2 +
\theta^1_3 \wedge \Theta^1_3 + \theta^2_3 \wedge \Theta^2_3
\bigr\}
\end{equation}__
Much to the surprise of Chern and Simons, physicists in
superconductivity and super-string theory both embraced
almost immediately the *Chern–Simons
action* defined by the 3-form __\( TP_1(\Theta^i_j) \)__.
Taken by itself, it is a closed 3-form which can be defined for any
connection on __\( M \)__ without any reference to a metric,
and it has continued to play an important role in
theoretical physics. This is a dramatic confirmation of Chern’s belief
in the importance of the forms themselves.

The Chern–Simons invariants cannot be defined unless we have a Pontryagin
*form* equal to 0. This naturally raises the question of whether on
a given manifold with a vanishing Pontryagin class, there is a Riemannian
metric whose corresponding Pontryagin form is zero.

One more major piece of work that Chern did in his Berkeley years should not go unmentioned. In 1974, he and Jürgen Moser wrote a paper in a completely different direction [8]. Generalizing Élie Cartan’s work on real hypersurfaces of complex Euclidean space of dimension two, they defined what we now call the Chern–Moser invariants of such hypersurfaces in all dimensions. These invariants are a complete set of local invariants in the real analytic case. The study of these invariants is now a fundamental part of geometric complex analysis. Finally, in 1992, when he was already eighty, he found inspiration in his own work in the late forties and, with D. Bao and Z. Shen, made a strong advocacy for generalizing classical Riemannian geometry to the Finsler setting. This advocacy has attracted a following.

During his Berkeley years, his leadership was felt in other areas too, but none more so than in the founding of two mathematics institutes. In 1981, the proposal he made jointly with Calvin Moore and I. M. Singer to establish an institute in mathematics on the Berkeley campus was officially approved by the government, and the Mathematics Sciences Research Institute (MSRI) was born. Chern served as its first director until 1984. The operational model of MSRI differs significantly from the most eminent research institute of our time, the Princeton Institute for Advanced Study. In contrast with the latter, MSRI has no permanent faculty, and each year its activities are organized around clearly defined mathematical topics. Senior mathematicians in each topic area are invited to visit MSRI for (part of) the year to help organize the scientific activities. This model has been followed around the world by other institutes since then.

Starting in the seventies, Chern took the lead in re-establishing mathematical communications between the U.S. and China. After his official retirement from the University in 1979, his visits to China became more frequent. Given that China has venerated scholarship for three thousand years, it was easy for someone with Chern’s diplomatic skills and preeminence to function smoothly at the highest political level in China. This may partially explain how he was able to establish, in 1984, a mathematics research institute in his alma mater, Nankai University, in Tianjin. A main goal of the Nankai Institute has been to attract leading mathematicians around the world to visit Tianjin and make it an active center of mathematics. Chern pursued this goal with vigor, and the Chinese government did its share in making foreign visitors welcome. When Chern finally returned to China for good in 1999, the well-being of the institute became his final project. He made ambitious plans that were only partially realized at the time of his death.

Chern is survived by his son Paul L. Chern, daughter May P. Chu, and four grandchildren, Melissa, Theresa, Claire, and Albert. His wife of sixty years, Shih-Ning, passed away earlier in year 2000 in Tianjin.