Celebratio Mathematica

Shiing-Shen Chern

Shiing-shen Chern: 1911–2004

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 Al­lendo­er­ferWeil [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.


[1] article S.-S. Chern: “On in­teg­ral geo­metry in Klein spaces,” Ann. Math. (2) 43 : 1 (January 1942), pp. 178–​189. MR 0006075 Zbl 0147.​22303

[2] article S.-S. Chern: “The geo­metry of iso­trop­ic sur­faces,” Ann. Math. (2) 43 : 3 (July 1942), pp. 545–​559. MR 0006477 Zbl 0060.​39207

[3] article S.-S. Chern: “A simple in­trins­ic proof of the Gauss–Bon­net for­mula for closed Rieman­ni­an man­i­folds,” Ann. Math. (2) 45 : 4 (October 1944), pp. 747–​752. MR 0011027 Zbl 0060.​38103

[4] article S.-S. Chern: “Some new view­points in dif­fer­en­tial geo­metry in the large,” Bull. Am. Math. Soc. 52 : 1 (1946), pp. 1–​30. MR 0021706 Zbl 0063.​00834

[5] article S.-S. Chern: “Char­ac­ter­ist­ic classes of Her­mitian man­i­folds,” Ann. Math. (2) 47 : 1 (January 1946), pp. 85–​121. MR 0015793 Zbl 0060.​41416

[6] article S.-S. Chern: “The in­teg­rated form of the first main the­or­em for com­plex ana­lyt­ic map­pings in sev­er­al com­plex vari­ables,” Ann. Math. (2) 71 : 3 (May 1960), pp. 536–​551. MR 0125979 Zbl 0142.​04802

[7]R. Bott and S. S. Chern: “Her­mitian vec­tor bundles and the equidistri­bu­tion of the zer­oes of their holo­morph­ic sec­tions,” Acta Math. 114 : 1 (1965), pp. 71–​112. A Rus­si­an trans­la­tion was pub­lished in Matem­atika 14:2 (1970). MR 0185607 Zbl 0148.​31906 article

[8] article S.-S. Chern and J. K. Moser: “Real hy­per­sur­faces in com­plex man­i­folds,” Acta Math. 133 : 1 (1974), pp. 219–​271. MR 0425155 Zbl 0302.​32015

[9] S.-S. Chern and J. Si­mons: “Char­ac­ter­ist­ic forms and geo­met­ric in­vari­ants,” Ann. Math. (2) 99 : 1 (January 1974), pp. 48–​69. MR 0353327 Zbl 0283.​53036 article