Celebratio Mathematica

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Thirty-six years to­geth­er
Through times of hap­pi­ness
And times of worry too.
Time’s pas­sage has no mercy.

We fly the Skies and cross the Oceans
To ful­fill my des­tiny;
Rais­ing the chil­dren fell
En­tirely on your shoulders.

How for­tu­nate I am
To have my works to look back upon,
I feel re­grets you still have chores.

Grow­ing old to­geth­er in El Cer­rito is a bless­ing.
Time passes by,
And we hardly no­tice.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

• Geo­met­ric struc­tures and their equi­val­ence prob­lems

• In­teg­ral geo­metry

• Eu­c­lidean dif­fer­en­tial geo­metry

• Min­im­al sur­faces and min­im­al sub­man­i­folds

• Holo­morph­ic maps

• Webs

• Ex­ter­i­or Dif­fer­en­tial Sys­tems and Par­tial Dif­fer­en­tial Equa­tions

• The Gauss–Bon­net The­or­em

• Char­ac­ter­ist­ic classes

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

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

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

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

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

1. a sub­group $G$ of $\mathbf{GL}(n,\mathbf{R})$,

2. cer­tain loc­al coframe fields $\{{\theta }_i\}$ in open sub­sets $U$ of ${\mathbf{R}}^n$ (i.e., $n$ lin­early in­de­pend­ent dif­fer­en­tial 1-forms in $U$).

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

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

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

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

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

Us­ing cur­rent geo­met­ric ter­min­o­logy, a $G$-struc­ture for a smooth n-di­men­sion­al man­i­fold $M$ is a re­duc­tion of the struc­ture group of its prin­cip­al tan­gent coframe bundle from $\mathbf{GL}(n,\mathbf{R})$ to the sub­group $G$. In par­tic­u­lar, the total space of this re­duc­tion is a prin­cip­al $G$-bundle, $P$, over $M$ con­sist­ing of the ad­miss­ible coframes $$\theta =({\theta }_1,\dots ,{\theta }_n),$$ and we can identi­fy the $G$-struc­ture with this $P$. There are $n$ ca­non­ic­ally defined 1-forms ${\omega }_i$ on $P$; if $$\mathrm{\Pi }:P\to M$$ is the bundle pro­jec­tion, then the value of ${\omega }_i$ at $\theta$ is ${\mathrm{\Pi }}^{\ast }({\theta }_i)$. The ker­nel of $D\mathrm{\Pi }$ is of course the sub­bundle of the tan­gent bundle $TP$ of $P$ tan­gent to its fibers, and is usu­ally called the ver­tic­al sub­bundle $V$. Clearly the ca­non­ic­al forms ${\omega }_i$ van­ish on $V$. The group $G$ acts on the right on $P$, act­ing simply trans­it­ively on each fiber, so we can identi­fy the ver­tic­al space $V_{\theta }$ at any point $\theta$ with the Lie al­gebra $L(G)$ of left-in­vari­ant vec­tor fields on $G$. Now, as Ehresmann first noted, a “con­nec­tion” in Cartan’s sense for the giv­en $G$-struc­ture (or as we now say, a $G$-con­nec­tion for the prin­cip­al bundle $P$) is the same as a “ho­ri­zont­al” sub­bundle $H$ of $TP$ com­ple­ment­ary to $V$ and in­vari­ant un­der $G$. In­stead of $H$ it is equi­val­ent to con­sider the pro­jec­tion of $TP$ onto $V$ along $H$ which, by the above iden­ti­fic­a­tion of $V_{\theta }$ with $L(G)$, is an $L(G)$-val­ued 1-form $\omega$ on $P$, called the “con­nec­tion 1-form”. If we de­note the right ac­tion of $g\in G$ on $P$ by $R_g$, then the in­vari­ance of $H$ un­der $G$ trans­lates to the trans­form­a­tion law $$R_g^{\ast }(\omega )=\mathrm{ad}(g^{-1})\circ \omega$$ for $\omega$, where $\mathrm{ad}$ de­notes the ad­joint rep­res­ent­a­tion of $G$ on $L(G)$. $L(G)$-val­ued forms on $P$ trans­form­ing in this way are called equivari­ant. Since $L(G)$ is a sub­al­gebra of the Lie al­gebra $L(\mathbf{GL}(n,\mathbf{R}))$ of $n\times n$ matrices, we can re­gard $\omega$ as an $n\times n$ mat­rix-val­ued 1-form on $P$, or equi­val­ently as a mat­rix ${\omega }_{ij}$ of $n^2$ real-val­ued 1-forms on $P$.

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

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

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

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

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

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

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

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

Chern first ex­ten­ded the clas­sic­al no­tion of “in­cid­ence” to a pair of ele­ments from two ho­mo­gen­eous spaces $G/H$ and $G/K$ of the same group $G$. Giv­en $aH\in G/H$ and $bK\in G/K$, Chern calls them “in­cid­ent” if $$aH\cap bK\ne \mathrm{\varnothing }.$$ This defin­i­tion plays an im­port­ant role in the the­ory of Tits build­ings.

In [13] and [17] Chern ob­tained fun­da­ment­al kin­emat­ic for­mu­las for two sub­man­i­folds in ${\mathbf{R}}^n$. The in­teg­ral in­vari­ants in Chern’s for­mula arise nat­ur­ally in Weyl’s for­mula for the volume of a tube $T_{\rho }$ of ra­di­us $\rho$ about a $k$-di­men­sion­al sub­man­i­fold $X$ of ${\mathbf{R}}^n$. Set­ting $m=n-k$, Weyl’s for­mula is: $$V(T_{\rho })=\sum _{0\le i\le k,i\text{ even}}{c}_i{\mu }_i(X){\rho }^{m+i}.$$ Here the $c_i$ are con­stants de­pend­ing on $m$ and $i$, $${\mu }_i(X)={\int }_M{I}_i(\mathrm{\Omega })$$ where $I_i$ is a cer­tain ad­joint in­vari­ant poly­no­mi­al of de­gree $i/2$ on the Lie al­gebra of $\mathbf{O}(n)$, and $\mathrm{\Omega }$ is the curvature form with re­spect to the in­duced met­ric on $X$. Chern’s for­mula (also dis­covered in­de­pend­ently by Fe­der­er) is: $$\int {\mu }_e(M_1\cap g{M}_2)dg=\sum _{0\le i\le e,i\text{ even}}{c}_i{\mu }_i(M_1){\mu }_{e-i}(M_2),$$ where $M_1$ and $M_2$ are sub­man­i­folds of ${\mathbf{R}}^n$ of di­men­sions $p$ and $q$ re­spect­ively, $e$ is even, $0\le e$ $\le p+q-n$, and $c_i$ are con­stant de­pend on $n,p,q,e$. Grif­fiths made the fol­low­ing com­ment con­cern­ing this pa­per [e3]:

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

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

1. $I$ is the in­duced met­ric on $M$,

2. $\mathit{II}$ is a quad­rat­ic form on $M$ with val­ues in the nor­mal bundle $\nu (M)$ such that, for any unit tan­gent vec­tor $u$ and unit nor­mal vec­tor $v$ at $p$, $${\mathit{II}}_v(u)=⟨\mathit{II}(u),v⟩$$ is the curvature at $p$ of the plane curve $\sigma$ formed by in­ter­sect­ing $M$ with the plane spanned by $u$ and $v$, and

3. if $s$ is a smooth nor­mal field then ${\mathrm{\nabla }}^{\nu }(s)$ is the or­tho­gon­al pro­jec­tion of the dif­fer­en­tial $ds$ onto the nor­mal bundle $\nu (M)$.

${\mathit{II}}_v=⟨\mathit{II},v⟩$ is called the second fun­da­ment­al form in the dir­ec­tion of $v$. The self-ad­joint op­er­at­or $A_v$ cor­res­pond­ing to ${\mathit{II}}_v$ is called the shape op­er­at­or of $M$ in the dir­ec­tion $v$.

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

(1)  Min­im­al sur­faces

Since the first vari­ation for the area func­tion­al for sub­man­i­folds of ${\mathbf{R}}^n$ is the trace of the second fun­da­ment­al form, a sub­man­i­fold $M$ of ${\mathbf{R}}^n$ is called min­im­al if $$\mathrm{trace}(\mathit{II})=0.$$ Let $\mathbf{Gr}(2,n)$ de­note the Grass­mann man­i­fold of 2-planes in ${\mathbf{R}}^n$. The Gauss map $G$ of a sur­face $M$ in ${\mathbf{R}}^n$ is the map from $M$ to the Grass­mann man­i­fold $\mathbf{Gr}(2,n)$ defined by $$G(x)=\text{the tangent plane to }M\text{ at }x.$$ The Grass­mann man­i­fold $\mathbf{Gr}(2,n)$ can be iden­ti­fied as the hy­per­quad­ric $$z_1^2+\cdots +z_n^2=0$$ of ${\mathbf{CP}}^{n-1}$ (via the map that sends a 2-plane $V$ of ${\mathbf{R}}^n$ to the com­plex line spanned by $e_1+i{e}_2$, where $(e_1,e_2)$ is an or­thonor­mal base for $V$). Thus $\mathbf{Gr}(2,n)$ has a com­plex struc­ture. On the oth­er hand, an ori­ented sur­face in ${\mathbf{R}}^n$ has a con­form­al and hence com­plex struc­ture through its in­duced Rieman­ni­an met­ric. In [16], Chern proved that an im­mersed sur­face in ${\mathbf{R}}^n$ is min­im­al if and only if the Gauss map is an­ti­ho­lo­morph­ic. This the­or­em was proved by Pinl for $n=4$ and is the start­ing point for re­lat­ing min­im­al sur­faces with the value dis­tri­bu­tion the­ory of Nevan­linna, Weyl, and Ahlfors. One of the fun­da­ment­al res­ults of min­im­al sur­face the­ory is the Bern­stein unique­ness the­or­em, which says that a min­im­al graph $z=f(x,y)$ in ${\mathbf{R}}^3$, defined for all $(x,y)\in {\mathbf{R}}^2$, must be a plane. Note that the im­age of the Gauss map of an en­tire graph lies in a hemi­sphere. Bern­stein’s the­or­em as gen­er­al­ized by Os­ser­man says that if the im­age of the Gauss map of a com­plete min­im­al sur­face of ${\mathbf{R}}^3$ is not dense, then the min­im­al sur­face is a plane. In [16], us­ing a clas­sic­al the­or­em of E. Borel, Chern gen­er­al­ized the Bern­stein–Os­ser­man the­or­em to a dens­ity the­or­em on the im­age of the Gauss map of a com­plete min­im­al sur­face in ${\mathbf{R}}^n$, that is not a plane. More re­fined dens­ity the­or­ems were es­tab­lished in [18], a joint pa­per with Os­ser­man.

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

(2)  Tight and taut im­mer­sions

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

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

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

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

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

When $n=2$, the Lie al­gebra $L(\mathbf{SO}(n))$ is 1-di­men­sion­al; ${\omega }_{11}={\omega }_{22}=0$ and ${\omega }_{21}=-{\omega }_{12}$, so there is only one in­de­pend­ent ${\omega }_{ij}$, namely ${\omega }_{12}$, and so only one curvature equa­tion, $$d{\omega }_{12}=-{\mathrm{\Omega }}_{12}=-R_{1212}{\omega }_1\wedge {\omega }_2.$$ Now it is eas­ily seen that $R_{1212}$ is a con­stant on every fiber ${\mathrm{\Pi }}^{-1}(x)$, and its value is in fact the Gaus­si­an curvature $K(x)$. We can identi­fy the area 2-form, $\mathit{dA}$, on $M$ with $-{\theta }_1\wedge {\theta }_2$, where $({\theta }_1,{\theta }_2)$ is any ori­ented or­thonor­mal frame, so that $${\mathrm{\Pi }}^{\ast }(\mathit{dA})=-{\omega }_1\wedge {\omega }_2.$$ Thus we can re­write the above curvature equa­tion as a for­mula for the pull-back of the Gauss–Bon­net in­teg­rand, $K\mathit{dA}$, to $F(M)$: $$\begin{array}{}\text{(*)}& {\mathrm{\Pi }}^{\ast }(K\mathit{dA})=d{\omega }_{12}.\end{array}$$ In [25] Chern re­marks that, along with zero tor­sion equa­tions, the for­mula $(\text{*})$ con­tains

“… all the in­form­a­tion on loc­al Rieman­ni­an geo­metry in two di­men­sions [and] gives glob­al con­sequences as well. A little med­it­a­tion con­vinces one that $(\text{*})$ must be the form­al basis of the Gauss–Bon­net for­mula, and this is in­deed the case. It turns out that the proof of the n-di­men­sion­al Gauss–Bon­net for­mula can be based on this idea….”

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

By ele­ment­ary to­po­logy, in the com­ple­ment $M^{\prime }$ of any point $p$ of a closed Rieman­ni­an man­i­fold $M$ one can al­ways define a smooth vec­tor field $e_1$ of unit length, and the in­dex of this vec­tor field at $p$ is $\chi (M)$. We will now see how this well-known char­ac­ter­iz­a­tion of the Euler char­ac­ter­ist­ic to­geth­er with the trans­gres­sion for­mula $(\text{*})$ leads quickly to Chern’s proof of the Gauss–Bon­net the­or­em for two-di­men­sion­al $M$. Let $e_2$ de­note the unit length vec­tor field in $M^{\prime }$ mak­ing $(e_1,e_2)$ an ori­ented frame, and let $\theta$ de­note the dual coframe field in $M^{\prime }$. Since $\mathrm{\Pi }$ com­posed with $\theta$ is the iden­tity map of $M^{\prime }$, we have $$d({\theta }^{\ast }({\omega }_{12}))={\theta }^{\ast }(d{\omega }_{12})=K\mathit{dA}$$ in $M^{\prime }$, so $${\int }_{M}K\mathit{dA}={\int }_{M^{\prime }}K\mathit{dA}={\int }_{M^{\prime }}d({\theta }^{\ast }({\omega }_{12})).$$ If we write $M_{\epsilon }$ for the com­ple­ment of the open $\epsilon$-ball about $p$, then $${\int }_{M^{\prime }}=\underset{\epsilon \to 0}{lim}{\int }_{M_{\epsilon }},$$ and by Stokes’ The­or­em, $${\int }_{M}K\mathit{dA}=\underset{\epsilon \to 0}{lim}{\int }_{S_{\epsilon }}{\theta }^{\ast }({\omega }_{12}),$$ where $S_{\epsilon }=\partial M_{\epsilon }$ is the dis­tance sphere of ra­di­us $\epsilon$ about $p$. The proof will be com­plete if we can identi­fy the right hand side of the lat­ter equa­tion with $2\pi$ times the in­dex of ${\epsilon }_1$ at $p$.

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

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

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

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

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

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

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

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

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

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

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

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

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

We will con­sider only the case of a Lie group $G$. Since the the­ory is es­sen­tially the same for a Lie group and one of its max­im­al com­pact sub­groups, we will also as­sume that $G$ is com­pact. A “space” will mean a para­com­pact to­po­lo­gic­al space, and a $G$-space will mean a space, $P$, to­geth­er with a con­tinu­ous right ac­tion of $G$ on $P$. We will write $R_g$ for the homeo­morph­ism $p\mapsto pg$. The $G$-space $P$ is called a prin­cip­al $G$-bundle if the ac­tion is free, i.e., if for all $p$ in $P$, $R_g(p)\ne p$ un­less $g$ is the iden­tity ele­ment $e$ of $G$. More spe­cific­ally, $P$ is called a prin­cip­al $G$-bundle over a space $X$ if we are giv­en some fixed homeo­morph­ism of $X$ with the or­bit space $P/G$, or equi­val­ently if there is giv­en a “pro­jec­tion map” $$\mathrm{\Pi }:P\to X$$ such that the $G$ or­bits of $P$ are ex­actly the “fibers” ${\mathrm{\Pi }}^{-1}(x)$ of the map $\mathrm{\Pi }$. $P$ is called the total space of the bundle, and we of­ten de­note the bundle by the same sym­bol as the total space. A map $\sigma :X\to P$ that is a left in­verse to $\mathrm{\Pi }$ is called a sec­tion. Two $G$-bundles over $X$, ${\mathrm{\Pi }}_i:P_i\to X$, $i=1,2$ are con­sidered “equi­val­ent” if there is a $G$-equivari­ant homeo­morph­ism $$\phi :P_1\to P_2\text{such that}{\mathrm{\Pi }}_1={\mathrm{\Pi }}_2\circ \phi .$$ The prin­cip­al $G$-bundle over $X$ defined by $P=X\times G$ with $R_g(x,\gamma )=(x,\gamma g)$ and $\mathrm{\Pi }(x,\gamma )=x$ is called the product bundle, and any bundle equi­val­ent to the product bundle is called a trivi­al bundle. Clearly $x\mapsto (x,e)$ is a sec­tion of the product bundle, so any trivi­al bundle has a sec­tion. Con­versely, if $\mathrm{\Pi }:P\to X$ has a sec­tion $\sigma$, then $\phi (x,g)=R_g(\sigma (x))$ is an equi­val­ence of the product bundle with $P$, i.e., a prin­cip­al $G$-bundle is trivi­al if and only if it ad­mits a sec­tion. We will de­note the set of equi­val­ence classes [P ] of prin­cip­al $G$-bundles $P$ over $X$ by ${\mathrm{Bndl}}_G(X)$.

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

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

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

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

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

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

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

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

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

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

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