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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 e1,,e2n be chosen (i.e., the ei are loc­ally defined vec­tor fields and are or­thonor­mal with re­spect to the met­ric). Let {ωi} be the dual co-frame field of {ei} (i.e., the ωi are 1-forms defined in the same neigh­bor­hood as the ei and ωi(ej)=δji for i,j=1,,2n). Fur­ther­more, let Ωji be the curvature form with re­spect to {ei}. Note that if e~1,,e~2n is an­oth­er frame field and h is the or­tho­gon­al trans­ition mat­rix between the frame fields, (1)e~i=kekhik, then the curvature form Ω~ji with re­spect to {e~i} sat­is­fies Ω~ji=,k(h1)iΩkhjk But (h1)i=hi, there­fore we have (2)Ω~ji=,khiΩkhjk Define the fol­low­ing 2n-form Ω by (3)Ω=122nπnn!εi1i2nΩi2i1Ωi2ni2n1 where εi1i2n is +1 or 1, de­pend­ing on wheth­er i1i2n is an even or odd per­muta­tion of 1,,2n, and is oth­er­wise equal to 0. This Ω is, a pri­ori, de­pend­ent on the choice of {ei} and is there­fore defined only in the neigh­bor­hood where {ei} is defined. Equa­tion (2), however, im­plies that if {ei} is re­placed by {e~i}, so that Ωji is re­placed by the curvature form Ω~ji cor­res­pond­ing to {e~i}, then εi1i2nΩi2i1Ωi2ni2n1=εi1i2nΩ~i2i1Ω~i2ni2n1. There­fore the form Ω is in­de­pend­ent of the the choice of the frame field {ei} 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 (4)MΩ=χ(M) where χ(M) de­notes the Euler char­ac­ter­ist­ic of M. We shall refer to Ω 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 F(M)={(x,f1,,f2n):xM, and f1,,f2n are(5)an orthonormal basis in the tangent space of M at x}. We have the pro­jec­tion map π:F(M)M. Since Ω is a form on M, the pull-back πΩ is a form on F(M). The ma­jor step of Chern’s proof of the Gauss–Bon­net the­or­em is that there ex­ists a (2n1)-form Π on F(M) so that (6)πΩ=dΠ 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 (6) is quite the op­pos­ite. Let θji be the con­nec­tion form of the Levi-Civ­ita con­nec­tion on F(M); θji (i,j=1,,2n) is a 1-form with value in the skew-sym­met­ric matrices so(2n), the Lie al­gebra of the spe­cial or­tho­gon­al group SO(2n). The curvature form Θji on F(M), a 2-form also tak­ing value in so(2n), is giv­en by Θji=dθji+kθkiθjk. This Θji is re­lated to the pre­ced­ing Ωji as fol­lows: If {ei} is the loc­al frame field as be­fore and Ωji is the curvature form with re­spect to {ei}, let e be the loc­al cross-sec­tion of π:F(M)M defined by e(x)=(x,e1(x),e2n(x)). Then e(Θji)=Ωji for all i, j.

The last fact about Θji and Ωji has the fol­low­ing con­sequence. Con­sider the 2n-form Θ defined on F(M) by (7)Θ=122nπnn!εi1i2nΘi2i1Θi2ni2n1 where εi1i2n has the same mean­ing as be­fore. It fol­lows from (3) that eΘ=Ω, so that (eπ)Θ=πΩ. A simple reas­on­ing2 shows that (eπ)Θ=Θ. Com­bin­ing these two re­la­tions, we get: (8)πΩ=Θ Thus to prove (6), it suf­fices to prove (9)Θ=dΠ for some (2n1)-form Π on F(M).

Chern’s proof of (9) re­quires the in­tro­duc­tion, on F(M), of the fol­low­ing (2n1)-forms Φ0, Φ1, …, Φn1, and the 2n-forms Ψ0, Ψ1, …, Ψn1: for each k=0,n1, Φk=εi1,,i2n1Θi2i1Θi2ki2k1θi2ni2k+1θi2ni2n1 and Ψk=(2k+1)εi1,,i2n1Θi2i1Θi2ki2k1Θi2ni2k+1θi2ni2k+2θi2ni2n1 where each sum is over all per­muta­tions i1,,i2n1 of 1, …, 2n1, and εi1,,i2n1 is equal to +1 or 1, de­pend­ing on wheth­er the per­muta­tion is even or odd. Note that from (7), we have (10)Ψn1=(22nπnn!)Θ Us­ing the Bi­an­chi iden­tity and the defin­i­tion of Θji in terms of the con­nec­tion form θji, one ob­tains the fol­low­ing re­cur­rence re­la­tion: (11)dΦk=Ψk1+2n2k12(k+1)Ψk where k=0,,n1 and Ψ10 by defin­i­tion. The sought-after (2n1)-form Π on F(M) is now defined to be (12)Π=1πnk=0n11135(2n2k1)2n+kk!Φk Us­ing (11), and then (10), we fi­nally get dΠ=122nπnn!Ψn1=Θ, which is ex­actly (9).

This proof of (9) 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 (4), we have to bring in the sphere bundle S(M)={(x,f):f is a unit vector in the tangent space of M at x}. F(M) is a fibre bundle over S(M) and we have a nat­ur­al pro­jec­tion π1:F(M)S(M). Briefly, the forms Φk and Ψk ac­tu­ally des­cend to S(M), in the sense that they are the pull-backs of forms in S(M) by π1. The same is there­fore true of Π and Θ, so that we may re­gard (9) as a re­la­tion between forms on S(M). Now giv­en any point x0 in M, the Hopf the­or­em on vec­tor fields says there is a unit vec­tor field v defined in M{x0} so that its isol­ated sin­gu­lar­ity at x0 has in­dex equal to χ(M), the Euler char­ac­ter­ist­ic of M. Re­gard­ing v as a cross-sec­tion of the bundle S(M)M over M{x0}, it is ele­ment­ary to see that, on M{x0}, Ω=d(vΠ) Moreover, and this is a crit­ic­al ob­ser­va­tion due to Chern, the re­stric­tion of Π (as a form on S(M)) to the fibre Sx0 of S(M) over x0 is ex­actly ΠSx0=(n1)!2πndσ where dσ is the volume form of the unit sphere Sx0 in the tan­gent space of M at x0.4 By a stand­ard ar­gu­ment, ex­press­ing M{x0} as the lim­it of M minus the small ball of ra­di­us ε around x0 as ε0 and us­ing Stokes’ the­or­em, the in­teg­ral of (4) be­comes the in­teg­ral Sx0vΠ=v(Sx0)Π=χ(M) where we have made use of the clas­sic­al fact that the volume of the unit sphere in 2n-di­men­sion­al space is 2πn/(n1)!. Now we see ex­actly why the Gauss–Bon­net the­or­em is true.

We may in­ter­pret the pre­ced­ing proof in the fol­low­ing way. The form Ω, 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 (4) 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 (6), 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 (6)), 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 Ωji (i,j=1,n) be the curvature form of the Her­mitian con­nec­tion re­l­at­ive to a loc­al unit­ary frame field {ei} (i=1,n) (i.e., each ei is a vec­tor field of type (1,0), and {ei(x)} is an or­thonor­mal basis of the holo­morph­ic tan­gent space at x for each x with re­spect to the Her­mitian met­ric). Ωji is of type (1,1). Con­sider now the fol­low­ing n dif­fer­en­tial forms ck(Ω) (k=1,n) of type (k,k): (13)ck(Ω)=(12π)ki1<i2<<ikσε(σ)Ωσ(i1)i1Ωσ(ik)ik where each σ ranges through all per­muta­tions of i1, …, ik, and the cor­res­pond­ing ε(σ) is the sign of the per­muta­tion, which is +1 if σ is even, and 1 if σ is odd. These are the Chern forms of the her­mitian met­ric. One ar­gues as in (3) above that these ck(Ω) do not de­pend on the choice of the unit­ary frame field {ei} so that they are glob­ally defined dif­fer­en­tial forms on M. A com­pu­ta­tion us­ing the Bi­an­chi iden­tity shows that in fact each ck(Ω) is a closed form. By de Rham’s the­or­em, each ck(Ω) rep­res­ents a co­homo­logy class of de­gree 2k, the k-th Chern class of the man­i­fold M. The un­usu­al look­ing coef­fi­cient in (13) guar­an­tees that the Chern classes are in­teg­ral classes.

Now let U(M)={(x,u1,,un):xM, and u1,,un are (14)an orthonormal basis of the holomorphic tangent space of M at x}. We shall refer to U(M) as the bundle of unit­ary frames over M. We have the nat­ur­al pro­jec­tion π:U(M)M. The ana­logue of (6) is that each of these forms ck(Ω), when pulled back to U(M), be­comes an ex­act form.5 As in (12), this fact is proved by an ex­pli­cit con­struc­tion: (15)πck(Ω)=d(Tck(Ω)) where each Tck(Ω) is a form ex­pli­citly con­struc­ted from ck(Ω). For sim­pli­city, we shall refer to Tck(Ω) as the trans­gres­sion of ck(Ω).6 Be­low, we shall have oc­ca­sion to refer to the fact that each Tck(Ω) can be writ­ten down ex­pli­citly in terms of ck(Ω) and the met­ric.

When the Her­mitian met­ric is Käh­leri­an, Chern iden­ti­fied the n-th Chern form cn(Ω) 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 ck(Ω) in (13) is based on the fact that the poly­no­mi­als cor­res­pond­ing to the ck(Ω) 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. (15)) in the top di­men­sion by prov­ing that, in case of an n-di­men­sion­al holo­morph­ic vec­tor bundle π:EM over an n-di­men­sion­al com­plex man­i­fold M, the pull-back of the top Chern form πcn(Ω) to E0 (here 0 stands for the zero sec­tion) is not only ex­act (see (15)), but “doubly ex­act”: πcn(Ω)=ddcρ for some (n1,n1) form ρ on E0. The non­tri­vi­al­ity of this as­ser­tion comes from the fact that E0 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(uji) be an in­vari­ant poly­no­mi­al on the Lie al­gebra so(n) of the or­tho­gon­al group O(n). If Θji is the curvature form on the frame bundle F(M) as in equa­tion (7) above, then P(Θji) is a closed from on F(M) that is the pull-back of a form on M (com­pare equa­tion (8)) and there­fore rep­res­ents a co­homo­logy class of M. Moreover, as in equa­tions (6) and (15), P(Θji) is ac­tu­ally an ex­act form on F(M), which we write as (16)P(Θji)=dTP(Θji) As men­tioned earli­er, the form TP(Θji) is ob­tained from P(Θji) by an ex­pli­cit con­struc­tion. \tex­tit{If the form P(Θji) is equal to 0}, then TP(Θji) 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(Θji) and TP(Θji) 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(Θji) does not change, and the co­homo­logy class of TP(Θji), as a class on F(M), does not change either. This co­homo­logy class [TP(Θji)] 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 P1 is the first Pontry­agin poly­no­mi­al, then P1(Θji), 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 [TP1(Θji)] on F(M) which is a con­form­al in­vari­ant of M. But in this case, the form TP1(Θji) can be simply writ­ten down: with θji as the con­nec­tion form on F(M) (see the nota­tion­al setup pre­ced­ing equa­tion (7)), (17)TP1(Θji)=14π2{θ21θ31θ32+θ21Θ21+θ31Θ31+θ32Θ32} 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 TP1(Θji). 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.

Works

[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