Chern, ShiingShen
CHERN, SHIINGSHEN
(b. Jia Xin, Chekiang Province, China, 26 October 1911;
d. Tianjin, China, 3 December 2004), mathematics, differential geometry.
Chern was a highly influential figure in pure mathematics. From the 1940s onward he redefined the subject of differential geometry by drawing on, and contributing to, the rapid development of topology during the period. Despite spending most of his working life in the United States, he was also a source of inspiration for all Chinese mathematicians, and contributed in many ways to the development of the subject in China.
Early Life . ShiingShen Chern was born in the final year of the Qing dynasty, and educated at a time when China was only beginning to set up Westernstyle universities. His father was a lawyer who worked for the government. Chern first showed his mathematical ability when he was a student at the Fu Luen Middle School in Tsientsin, where he did all the exercises in classical English textbooks on algebra and trigonometry. He then went to Nankai University at the age of fifteen. There, mathematics was a oneman department run by LiFu Chiang, who had been a student of Julian Coolidge, and this ensured that Chern studied a great deal of geometry, particularly the works of Coolidge, George Salmon, Guido Castelnuovo, and Otto Staude. He became a postgraduate in 1930 at Tsinghua University in Beijing, where he met his wife ShihNing, the daughter of a professor. At Tsinghua Chern came under the influence of Dan Sun, one of the few mathematicians in China publishing research in mathematics. Sun’s subject was projective differential geometry, which caught Chern’s interest, and he studied in detail the works of the German mathematician Wilhelm Blaschke. After Blaschke visited Tsinghua in 1932 and lectured on differentialgeometric invariants, Chern won a fellowship to study with him in Hamburg, Germany, for two years, and he received his DSc there in 1936 for work on the theory of webs, a subject central to Blaschke’s work at the time. These were turbulent times in Germany: in Hamburg Chern met the mathematician WeiLiang Chow, who had left Göttingen because of the flight of the best mathematicians from that university, and during the same period Blaschke was forced to resign from the German Mathematical Society for opposing the imposition of Nazi racial policies.
While in Hamburg, Chern studied the works of Elie Cartan and in 1936 spent a year in Paris with him. Cartan, who turned sixtyseven that year, was the dominant figure in geometry at the time, and had introduced new techniques that few people understood. The language in which to properly express Cartan’s work was not then available, and it was ten years before the notation and terminology of fiber bundles allowed Chern to explain these concepts in a satisfactory way. The regular “Séminaire Julia” that year was devoted to expounding Cartan’s work and Chern there met André Weil and other young French mathematicians who were the founders of the Bourbaki group that came to dominate French mathematics after World War II.
Move to the United States . In the summer of 1937 he took up the position of professor at Tsinghua, crossing the Atlantic, the United States, and the Pacific to do so, only to find that the SinoJapanese war had begun. His university had moved, with the universities of Peking and Nankai, to Kunming. There, despite all the deprivations of war and virtually cut off as he was from the outside world, he found the time to work deeply through Cartan’s work and came to his own vision of where geometry should be going. He was also able to teach many students who were to go on to make substantial contributions in mathematics and other fields—among them Chen Ning Yang, whose work in theoretical physics won him a Nobel Prize in 1957. Eventually, Chern was able to make his way to the Institute for Advanced Study in Princeton, New Jersey, on a series of military flights through India, Africa, Brazil, and Central America.
In Princeton, Hermann Weyl and Oswald Veblen already had a high opinion of Chern because of his papers. Chern soon got in touch with Claude Chevalley and Solomon Lefschetz and also with Weil in nearby Lehigh University. In Weil’s words, “We seemed to share a common attitude towards such subjects, or towards mathematics in general; we were both striving to strike at the root of each question while freeing our minds from preconceived notions about what others might have regarded as the right or the wrong way of dealing with it” (Weil, 1992, p. 74). Chern and Weil worked and talked together to reveal the topological character of some of the new ideas in algebraic geometry. These included the ToddEger classes, whose definition was at the time derived in the oldfashioned spirit of Italian geometry, but which nevertheless caught Chern’s imagination. These discussions provided the foundation of his most famous work on what became known as Chern classes (though he would always insist that the letter c by which they were denoted stood for “characteristic classes”). The ideas he developed at that time emerged in a concrete form in his new intrinsic proof (1944) of the general GaussBonnet theorem— by his own account, one of his favorite theorems.
When World War II ended in 1945, Chern began another lengthy, complicated return to China, reaching Shanghai in March 1946. There, he was asked to set up an institute of mathematics as part of the Academia Sinica. He did this very successfully—several outstanding mathematicians were nurtured there—but the institute was located in Nanjing, and the turmoil of the civil war was making southern China ever more dangerous. As a result, Weil, by then in Chicago, and Veblen and Weyl in Princeton became concerned about his fate, and both Chicago and Princeton’s Institute for Advanced Study offered Chern visiting positions, culminating in a full professorship at Chicago. So in 1949 he returned to the United States, this time with his family, to spend most of his working life there.
Chern’s topological interests in Nanjing and Chicago deepened as he absorbed the rapid postwar development in algebraic topology, and his talk at the 1950 International Congress of Mathematicians (1952) shows how dramatic the interaction of differential geometry and topology had become by then. It is a thoroughly modern statement, totally different in outlook from the work of fifteen years earlier.
Work in California and China . In 1960, Chern moved again, to become a professor at the University of California at Berkeley—attracted by an expanding department and a milder climate. There he immediately started a differential geometry seminar that continues in the early twentyfirst century, and he attracted visitors both young and old. His own PhD students included ShingTung Yau, who won a Fields Medal in 1982.
In 1978, the year he turned sixtyseven, Chern, Isadore Singer, and Calvin Moore prepared a response to the National Science Foundation’s request for proposals for a mathematical institute that would reflect the “need for continued stimulation of mathematical research” in an environment that considered American mathematics to be in a “golden age.” Their ideas were approved in 1981 and Chern became the first director of the Mathematical Sciences Research Institute, a post he held from 1982 until 1985. It was a huge success, and Chern supported it thereafter in many ways, not least from the proceeds of his 2004 Shaw Prize. A new building, Chern Hall, was dedicated in his memory on 3 March 2006.
Throughout his years in the United States Chern’s interest in Chinese mathematicians continued. He aimed to put Chinese mathematics on the same level as its Western counterpart, “though not necessarily bending its efforts in the same direction” (Citation, Honorary Doctorate, Hong Kong University of Science and Technology, 7 November 2003, available from http://genesis.ust.hk/jan_2004/en/camera/congregate/citations_txt05.html). During the 1980s, he initiated three developments in China: an International Conference on Differential Geometry and Differential Equations, the Summer Education Centre for Postgraduates in Mathematics, and the Chern Programme, aimed at helping Chinese postgraduates in mathematics to go for further study in the United States. In 1984 China’s Ministry of Education invited him
to return to his alma mater, Nankai University, and create the Nankai Research Institute of Mathematics. The university built a residence for him, “The Serene Garden,” and he and his wife lived there every time they returned to China. While director he invited many overseas mathematicians to visit; he also donated more than 10,000 books to the institute, and his $50,000 Wolf Prize to Nankai University.
In 1999 Chern returned to China for good, where he continued to do mathematics, grappling until just before his death with an old problem about the existence or otherwise of a complex structure on the sixdimensional sphere. The finest testament to his achievement in his final years was to be seated next to President Jiang Zemin in the Great Hall of the People in Beijing at the opening of the 2002 International Congress of Mathematicians. During the course of his lifetime, mathematics in China had changed immeasurably.
Chern received many awards for his work including the U.S. National Medal of Science in 1975, the Wolf Prize in Mathematics in 1983, and the Shaw Prize in 2004. He died on 3 December 2004 at age ninetythree; his wife of sixtyone years had died four years earlier. He was survived by a son, Paul, and a daughter May Chu.
Proof of the GaussBonnet Theorem . Chern’s mathematical work encompasses a period of rapid change in geometry, and he was exceptionally able to capitalize on his extensive knowledge of the mathematics of both the first and the second half of the twentieth century. His subject of differential geometry had its origins in the study of surfaces inside the threedimensional Euclidean space with which everyone is familiar. It involves the notions of the length of curves on the surface, the area of domains within it, the study of geodesics on the surface, and various concepts of curvature. By the late nineteenth century other types of geometry were being studied this way, such as projective geometry and web geometry, the subject on which Chern cut his mathematical teeth. An nweb in the plane consists of n families of nonintersecting curves that fill out a portion of the plane. For example, a curvilinear coordinate system such as planar Cartesian coordinates or polar coordinates defines a 2web. By a change of coordinates any planar 2web can be taken to the standard Cartesian system, but this is not so for webs of degree three and higher and invariants which have the nature of curvature obstruct this.
Most proofs related to the subject that appeared during this period involve intricate calculations, and Chern indeed was a master at such proofs. However, in the 1920s new inputs in differential geometry arrived from its importance in Einstein’s theory of general relativity. One of these was the shift in emphasis from twodimensional geometry to the fourdimensional geometry of spacetime. Coupled with the nineteenthcentury formulation of mechanics, which involved highdimensional configuration spaces where kinetic energy defined a similar structure to a surface in Euclidean space, the ruling perspective in differential geometry was to work in n dimensions. A second change brought on by relativity was the requirement that the equations of physics should be written in a coordinateindependent way. This required the introduction of mathematical objects that had a life of their own, but which could still be manipulated by indexed quantities so long as one knew the rules for changing from one coordinate system to another. The most fundamental change, however, was the movement from extrinsic geometry to intrinsic geometry: fourdimensional spacetime was not sitting like a surface in a higherdimensional Euclidean space; its geometry could be observed and described only by the beings that lived within it. The intrinsic viewpoint also paved the way for the global viewpoint—the spaces one needed to study, not least spacetime itself, could have quite complicated topology and one wanted to understand the interaction between the differential geometry and the topology: to see what constraints topology imposes on curvature, or viceversa.
This was the context of Chern’s proof of the general GaussBonnet theorem (1944), which was a pivotal event in the history of differential geometry, not just for the theorem itself but also for what it led to. The classical theorem of the same name concerns a closed surface in Euclidean threespace. It states that the integral of the Gaussian curvature is 2π times the Euler number. The Euler number for a surface divided into F faces, E edges and V vertices is VE+F. For a sphere this is 2, and the GaussBonnet theorem gives this because the Gaussian curvature of a sphere is 1, and its area is 4π.
This link between curvature and topology has several features: one is Gauss’s theorema egregium, which says that a certain expression of curvature of the surface, the Gauss curvature, is intrinsic—it can be determined by making measurements entirely within the surface. That being so, clearly whatever its integral evaluates to depends only on the intrinsic geometry. In contrast, there is a very natural and useful extrinsic interpretation of this integral as the degree of the Gauss map: the unit normal to the surface at each point defines a map to the sphere, and its topological degree (the number of points with the same normal direction) is the invariant. The problem was to extend this result to (evendimensional) manifolds in higher dimension. In 1926 Heinz Hopf had generalized the Gauss map approach to hypersurfaces in Euclidean nspace, but the task was to prove the theorem for any evendimensional Riemannian manifold. The concept of manifold, commonplace in mathematics today and signifying a higherdimensional analogue of a surface, was by no means clear when Chern was working on this theorem. Indeed, the definition was formulated correctly by Hassler Whitney only in 1936, and Cartan even in 1946 considered that “the general notion of manifold is quite difficult to define with precision” (Cartan, 1949, p. 56).
The novel content of the proof came from studying the intrinsic tangent sphere bundle, and using the exterior differential calculus that Chern had learned at the hands of Cartan. The language of fiber bundles was necessary to describe in an intrinsic way the totality of tangent vectors to a manifold—it was what Cartan lacked and was only developed amongst topologists in the period 1935–1940. Chern’s theorem, proved with the use of this concept, provided a link between topology and differential geometry at a time when the very basics of the topology of manifolds were being laid down.
Discovery of the Chern Classes . The successful attack on the GaussBonnet theorem led him to study the other invariants of bundles, to see whether curvature could detect them. He started with StiefelWhitney classes but their more algebraic properties “seemed to be a mystery” (Weil, 1992, p. 74), and what are now called Pontryagin classes, where curvature could make an impact, were not known then, so Chern moved into Hermitian geometry and discovered the famous Chern classes whose importance in algebraic geometry, topology, and index theory cannot be underestimated. As he pursued his work on characteristic classes and curvature, Chern always recognized that there was more than just the topological characteristic class to be obtained, and this emerged later in a strong form in his work on ChernSimons invariants with James Simons (1971). Nowadays the ChernSimons functional is an everyday tool for theoretical physicists.
The Chern classes, coupled with the Hodge theory that in the postwar period was given a more rigorous foundation by Kunihiko Kodaira and Weyl, provided a completely new insight into the interaction of algebraic geometry and topology. But Chern was always happy to work in algebraic geometry. His studies in Hamburg involved webs obtained from algebraic curves—a plane curve of degree d meets a general line in d points. There is a duality between points and lines in the plane: the oneparameter family of lines passing through a point describes a line in another plane. So the curve describes d families of lines, which is a web. Chern in fact later returned to this theme in far more generality in collaboration with the algebraic geometer Phillip Griffiths (1978). Nevertheless, it was the new differential and topological viewpoint on the traditional geometry in the complex domain that motivated most of his contributions. One of these was his work in several complex variables on value distribution theory. In joint work with Raoul Bott (1965) he introduced the use of connections and curvature on vector bundles into this area. In fact, their formulation of the notion of a connection in that paper is so simple and manageable that it has become the standard approach in the literature. In this context a vector bundle is a smooth family of abstract vector spaces parameterized by the points of a manifold (like the tangent spaces of a surface) and a connection is an invariant way of taking the derivative of a family of vectors.
Another link between the algebraic geometric and differential geometric world that Chern contributed to is in the area of minimal surfaces, the simplest examples of which are the surfaces formed by the soap films spanning a wire loop. Chern was the first to attempt a rigorous proof of what is classically known as the existence of isothermal coordinates on a surface. On any surface, such as a surface sitting in Euclidean space, one can find two real coordinates which are described by a single complex number. This immediately links the differential geometry of surfaces with complex analysis, and the most direct case is that of a minimal surface. The physicist Yang learned about this taking a course from Chern in China in 1940: “When Chern told me to use complex variables … it was like a bolt of lightning which I never later forgot” (Yang, 1992, p. 64). In later work, Chern discussed minimal surfaces in higher dimensional Euclidean spaces and in spheres and showed how in quite intricate ways the algebraic and differential geometry intertwine.
Other Mathematical Work . Chern’s work on characteristic classes earned him a large audience of mathematicians in a variety of disciplines, but he did not neglect the other aspects of differential geometry, especially where unconventional notions of curvature were involved. Some of this arose from early attempts to extend general relativity—for example, Weyl geometry and path geometry. The latter considers a space which has a distinguished family of curves on it that behave qualitatively like geodesics— given a point and a direction there is a unique curve of the family passing through the point and tangent to the direction. Veblen and his school in Princeton had worked on this and it was through this work that they probably first heard of Chern. Curvature invariants in complex geometry also came up in his work with Jürgen Moser (1974) on the geometry of real hypersurfaces in a complex vector space, picking up on a problem once considered by Cartan. When, in the mid1970s, soliton equations such as the KdV equation, together with its socalled Bäcklund transformations, began to be studied, he was well prepared to apply both his expertise in exterior differential systems and his knowledge of classical differential geometry to provide important results.
Sometimes his choice of topics was unorthodox, but reflected both his curiosity and respect for the mathematicians of the past. Bernhard Riemann in his famous inaugural lecture of 1854, On the Hypotheses which Lie at the Basis of Geometry, discussed various competing notions of infinitesimal length but concluded that it would “take considerable time and throw little new light on the theory of space, especially as the results cannot be geometrically expressed; I restrict myself therefore to those manifoldnesses in which the line element is expressed as the square root of a quadric differential expression” (Riemann, 1873, p. 17). His “restricted” theory is what is known as Riemannian geometry in the early twentyfirst century. The alternatives have come to be called Finsler metrics; Chern took Riemann at face value and set out with collaborators to investigate the geometry of these (2000).
In a life as long and full as Chern's, there are many more highly significant contributions. He also returned to some favorite themes over the decades. One was Blaschke’s use of integral geometry and generalizations of the attractive Crofton’s formula, which measures the length of a curve by the average number of intersections with a line. Despite his geometrical outlook, Chern’s proofs were usually achieved by the use of his favorite mathematical objects—differential forms. He had learned this skill with Cartan and was an acknowledged master at it.
One of the enduring features in Chern’s life was his accessibility and offers of encouragement to young mathematicians: As Bott remarked, “Chern treats people equally; the high and mighty can expect no courtesy from him that he would not also naturally extend to the lowliest among us” (Bott, 1992, p. 106). His relaxed style and willingness to help young researchers earned him loyalty from generations of mathematicians. One such appreciative student bought his weekly California State Lottery tickets with the single thought “If I win, I will endow a professorship to honor Professor Chern.” In 1995 he won $22 million and the Chern Visiting Professors became a regular feature on the Berkeley campus.
BIBLIOGRAPHY
WORKS BY CHERN
“A Simple Intrinsic Proof of the GaussBonnet Formula for Closed Riemannian Manifolds.” Annals of Mathematics 45 (1944): 747–752.
“On the Curvatura Integra in a Riemannian Manifold.” Annals of Mathematics 46 (1945): 674–684. “Differential Geometry of Fiber Bundles.” In Proceedings of the
International Congress of Mathematicians, Cambridge, Mass., 1950. Vol. 2. Providence, RI: American Mathematical Society, 1952.
“An Elementary Proof of the Existence of Isothermal Parameters on a Surface.” Proceedings of the American Mathematical Society6 (1955): 771–782.
With Richard Lashof. “On the Total Curvature of Immersed Manifolds.” American Journal of Mathematics 79 (1957): 306–318.
With Raoul Bott. “Hermitian Vector Bundles and the Equidistribution of the Zeroes of Their Holomorphic Sections.” Acta Mathematica 114 (1965): 71–112.
Complex Manifolds without Potential Theory. Princeton, NJ, Toronto, and London: Van Nostrand, 1967. 2nd edition, New York and Heidelberg: SpringerVerlag, 1979, and revised printing of the 2nd edition, New York: SpringerVerlag, 1995.
With James Simons. “Some Cohomology Classes in Principal Fiber Bundles and Their Application to Riemannian Geometry.” Proceedings of the National Academy of Sciences of the United States of America 68 (1971): 791–794.
With Jürgen K. Moser. “Real Hypersurfaces in Complex Manifolds.” Acta Mathematica 133 (1974): 219–271. Erratum: 150 (1983): 297.
With Phillip A. Griffiths. “An Inequality for the Rank of a Web and Webs of Maximum Rank.” Annali della scuola normale superiore di Pisa classe di scienze 5 (1978): 539–557.
With Jon G. Wolfson. “Harmonic Maps of the TwoSphere into a Complex Grassmann Manifold II.” Annals of Mathematics125 (1987): 301–335.
With David Bao and Zhongmin Shen. An Introduction to RiemannFinsler Geometry. New York: Springer, 2000.
OTHER SOURCES
Bott, Raoul. “For the Chern Volume.” In Chern, a Great Geometer of the Twentieth Century, edited by ShingTung Yau. Hong Kong: International Press, 1992.
Cartan, Elie. Leçons sur la Géométrie des Espaces de Riemann. 2nd ed. Paris: GauthierVillars, 1946.
Hitchin, Nigel J. “ShiingShen Chern 1911–2004.” Bulletin of the London Mathematical Society 38 (2006): 507–519. Contains a complete list of Chern’s works.
Jackson, Allyn. “Interview with Shiing Shen Chern.” Notices of the American Mathematical Society 45 (1998): 860–865.
Riemann, Bernhard. “On the Hypotheses which Lie at the Bases of Geometry.” Translated by William K. Clifford. Nature 8 (1873): 14–17, 36, 37.
Weil, André. “S. S. Chern as Geometer and Friend.” In Chern, a Great Geometer of the Twentieth Century, edited by ShingTung Yau. Hong Kong: International Press, 1992.
Yang, Chen Ning. “S. S. Chern and I.” In Chern, a Great Geometer of the Twentieth Century, edited by ShingTung Yau. Hong Kong: International Press, 1992.
Nigel J. Hitchin
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PINYIN → WADEGILES CONVERSION TABLE
PINYIN → WADE‐GILES CONVERSION TABLE
A‐luo‐ben  A‐lo‐pen 
An Shigao  An Shih‐kao 
Baduanjin  Pa‐tuan‐chin 
Bagua  Pa‐kua 
Bailianzong  Pai‐lien‐tsung 
Baimasi  Pai‐ma‐ssu 
Baiyunguan  Pai‐yün Kuan 
Baizhang Huaihai  Pai‐chang Huai‐hai 
Baizhangqinggui  Pai‐Chang‐Ch'ing‐Kuei 
Bajiao Huiqing  Pa‐chiao Hui‐ch'ing 
Baxian  Pa‐hsien 
Bigu  Pi‐ku 
Biqiu  Pi‐ch'iu 
Biyanlu  Pi‐yen‐lu 
Bukong Jingang  Pu‐k'ung Chin‐kang 
Caishen  Ts'ai‐shen 
Caodong  Ts'ao‐tung 
Cao Guojiu  Ts'ao Kuo‐chiu 
Caoshan Benji  Ts'ao‐shan Pen‐chi 
Chan  Ch'an 
Chang  Ch'ang 
Chang'an  Ch'ang‐an 
Changsha Jingcen  Ch'ang‐sha Ching‐ts'en 
Changsheng Busi  Ch'ang‐sheng Pu‐ssu 
Channa  Ch'an‐na 
Chanzong  Ch'an‐tsung 
Cheng  Ch'eng 
Cheng Hao  Ch'eng Hao 
Chenghuang  Ch'eng‐huang 
Chengshi  Ch'eng‐shih 
Cheng Yi  Ch'eng I 
Cheng Yi  Ch'eng Yi 
Chen Tuan  Ch'en T'uan 
Chenzhu  Ch'eng‐chu 
Chun Qiu  Ch'un Ch'iu 
Cunsi  Ts'un‐ssu 
Dacheng  Ta‐ch'eng 
Dachu Huihai  Ta‐chu Hui‐hai 
Dahui Zonggao  Ta‐hui Tsung‐kao 
Danxia Tianran  Tan‐hsia T'ien‐jan 
Dao  Tao 
Daoan  Tao‐an 
Daochuo  Tao‐ch'o 
Daodejing  Tao‐te Ching 
Daodetianzun  Tao‐te t'ien‐tsun 
Dao Hongjing  T'ao Hung‐ching 
Daojia  Tao‐chia 
Daojiao  Tao‐chiao 
Daosheng  Tao‐sheng 
Daoshi  Tao‐shih 
Daoyi  Tao‐i 
Daoyin  Tao‐yin 
Daozang  Tao‐tsang 
Datongshu  Ta T'ung Shu 
Daxue  Ta Hsüeh 
Deshan Xuanjian  Te‐shan Hsüan‐chien 
Di  Ti 
Dicang  Ti‐ts'ang 
Dongshan Liangjie  Tung‐shan Liang‐chieh 
Doushuai Congyue  Tou‐shuai Ts'ung‐yueh 
Dunhuang  Tun‐huang 
Dunwu rudao yaomen‐lun  Tun‐wu ju‐tao yao‐men lun 
Dushan  Tu‐shun 
Emei, Mount  O‐mei, Mount 
Emito, Mount  O‐mi‐t'o 
Fa  Fa 
Fajia  Fa‐chia 
Falang  Fa‐lang 
Fangshi  Fang‐shih 
Fang Yangou  Fang Yen‐kou 
Fangzhang  Fang‐chang 
Fangzhongshu  Fang‐chung shu 
Farong  Fa‐jung 
Fashun  Fa‐shun 
Faxian  Fa‐hsien 
Faxiang  Fa‐hsiang 
Fayan Wenyi  Fa‐yen Wen‐i 
Fazang  Fa‐tsang 
Fazhu  Fa‐ju 
Feisheng  Fei‐sheng 
Fenggan  Feng‐kan 
Fengshui  Feng‐shui 
Fenyang Shanzhao  Fen‐yang Shan‐chao 
Fulu (bai)  Fu‐lu (pai) 
Fuqi  Fu‐ch'i 
Ge Hong  Ko Hung 
Geyi  Ko‐yi 
Gong'an  Kung‐an 
Gu, Ku  Ku, K'u 
Gui  Kuei 
Guifeng Zongmi  Kuei‐feng Tsung‐mi 
Guishan Lingyu  Kuei‐shan Ling‐yu 
Guiyangzong  Kuei‐yang‐tsung 
Hanshan  Han‐shan 
Hanshu  Han Shu 
Han Xiangzi  Han Hsiang‐tzu 
Han Yu  Han Yü 
Heqi  Ho‐ch'i 
Heshang  Ho shang 
Heshanggong  Ho‐shang kung 
He Xiangu  Ho Hsien‐ku 
Hongren  Hung‐jen 
Hong Xiuchuan  Hung Hsiu‐ch'uan 
Hongzhi Zhengque  Hung‐chih Cheng‐ch'üeh 
Huahu Jing  Hua‐Hu Ching 
Huainanzi  Huai‐nan Tzu 
Huangbo Xiyun  Huang‐po Hsi‐yün 
Huangdi  Huang‐ti 
Huangdi neijing  Huang‐ti nei‐ching 
Huangjin  Huang‐chin 
Huanglaojun  Huang‐lao Chün 
Huanglong Huinan  Huang‐lung Hui‐nan 
Huangquan  Huang‐ch'üan 
Huangtingqing  Huang‐t'ing Ching 
Huanjing  Huan‐ching 
Hua Tuo  Hua T'o 
Huayan  Hua‐yen 
Hui  Hui 
Huichang  Hui‐ch'ang 
Huiguo  Hui‐kuo 
Huike  Hui‐k'o 
Huineng  Hui‐neng 
Hui Shi  Hui Shih 
Huiyuan  Hui‐yuan 
Huizong  Hui‐tsung 
Ji  Ki 
Jiang Yi  Chiang‐I 
Jianzhen  Chien‐chen 
Jiao  Chiao 
Jie  Kie 
Jindan  Chin‐tan 
Jing  Ching 
Jingde Quangdenglu  Ching‐te Ch'üan‐teng‐lu 
Jingru  Chinju 
Jingtun  Ching‐t'un 
Jinlian  Chin‐lien 
Junzi  Chün tzu 
Kong  K'ung 
Kongzi  K'ung‐tzu 
Kou Jianzhi  K'ou Chien'chih 
Kuiqi  K'uei‐chi 
Kunlun  K'un‐lun 
Lan Zaihe  Lan Tsai‐ho 
Lao‐Dan  Lao Tan 
Laozhun  Lao‐chun 
Laozi  Lao‐Tzu 
Li  Li 
Li Bai  Li Po 
Li Bai, Li Taibai  Li Pai, Li T'ai‐pai 
Lien‐ch'i  Liangi 
Lie Yukao  Lieh Yü‐k'au 
Liezi  Lieh‐Tzu 
Ligui  Li‐kuei 
Li Ji  Li Chi 
Linji Yixuan  Lin‐chi I‐hsüan 
Li Tieguai  Li T'ieh‐kuai 
Liuzidashi  Liu‐tsu‐ta‐shih 
Liuzidashi Fabaotanjing  Liu‐Tsu‐Ta‐Shih Fa‐Pao‐T'an‐Ching 
Lixue  Li‐Hsüeh 
Li Zixu  Li Tsu‐hsu 
Long  Lung 
Longhua  Lung‐hua 
Longhushan  Lung‐hu‐shan 
Longmen  Lung‐men 
Longtan Zhongxiu  Lung‐t'an Chung‐hsiu 
Longwang  Lung‐wang 
Lu  Lu 
Ludongbin  Lu tung‐pin 
Lun Yü  Lun Yü 
Luohan  Lo‐han 
Luo Qing  Lo Ch'ing 
Luoyang  L(u)oyang 
Lushan  Lu‐shan 
Lüshi Chunqiu  Lü‐shih Ch'un‐Ch'iu 
Lüzong  Lü‐tsung 
Mafa  Ma‐fa 
Mandalao  Man‐ta‐lao 
Maoshan  Mao‐shan 
Mao Ziyuan  Mao Tzu‐yuan 
Mazu  Ma‐tsu 
Mazu Daoyi  Ma‐tsu Tao‐i 
Mengzi  Meng Tzu 
Menshen  men‐shen 
Miluofo  Mi‐lo‐fo 
Mingdao  Ming‐tao 
Mingdi  Ming‐ti 
Ming Ji  Ming Chi 
Modi  Mo Ti 
Mogao Caves  Mo‐kao Caves 
Mojia  Mo‐chia 
Mozhao chan  Mo‐chao ch'an 
Mozi  Mo Tzu 
Nanhua zhenjing  Nan‐hua chen‐ching 
Nanyang Huizhong  Nan‐yang Hui‐chung 
Nanyuan Huiyong  Nan‐yuan Hui‐yung 
Nanyue Huairang  Nan‐yüeh Huai‐jang 
Neidan  Nei‐tan 
Neiguan  Nei‐kuan 
Neiqi  Nei‐ch'i 
Neishi  Nei‐shih 
Nianfo  Nien‐fo 
Nieban  Nieh‐pan 
Nügua  Nü‐kua 
Pangu  P'an‐ku 
Pang Yun  P'ang Yün 
Pangzhushi  P'ang‐chu‐shih 
Peixiu  P'ei Hsiu 
Peng Lai  P'eng Lai 
Pengzi  P'eng‐tzu 
Po  P'o 
Pu  P'u 
Puhua  P'u‐Hua 
Pusa  P'u‐sa 
Puteshan  P'u‐t'o‐shan 
Putidamo  P'u‐t'i‐ta‐mo 
Puxian  P'u‐hsien 
Qi  Ch'i 
Qian ai  Ch'ien ai 
Qian zi wen  Ch'ien tzu wen 
Qielan  Ch'ieh‐Ian 
Qigong  Ch'i‐kung 
Qihai  Ch'i‐hai 
Qin  Ch'in 
Qingming  Ch'ing‐ming 
Qingtan  Ch'ing‐t'an 
Quanzhendao  Ch'üan‐chen tao 
Rujia  Ju‐chia 
Rujia  Ru‐chia 
Rujiao  Ju‐chiao 
Rulai  Ju‐lai 
Sanbao  San‐pao 
Sanjia  San‐chiao 
Sanjiejiao  San‐chieh‐chiao 
Sanqing  San‐ch'ing 
Sanshen  San‐Shen 
Sansheng Huiran  San‐Sheng Hui‐Jan 
Sanxing  San‐hsing 
Sanyi  San‐i 
Shandao  Shan‐tao 
Shangdi  Shang‐ti 
Shangqing  Shang‐Ch'ing 
Shangqing  Shang‐Ch'ing 
Shangzuobu  Shang‐tso‐pu 
Shaolinsi  Shao‐lin‐ssu 
Shen  Shen 
Sheng(‐ren)  Sheng(‐jen) 
Shenhui  Shen‐hui 
Shenxiang  Shen‐hsiang 
Shenxiu  Shen‐hsiu 
Shi  Shih 
Shiji  Shih‐Chi 
Shijie  Shih‐chieh 
Shijing  Shih Ching 
Shishu  Shih‐shu 
Shishuang Chuyuan  Shih‐shuang Ch'u‐yuan 
Shiyi  Shih‐i 
Shou  Shou 
Shoulao  Shou‐lao 
Shoushan Shengnian  Shou‐shan Sheng‐nien 
Shouyi  Shou‐i 
Shu  Shu 
Shujing  Shu Ching 
Sima Qian  Ssu‐ma Ch'ien 
Siming  Ssu‐ming 
Sishu  Ssu Shu 
Sixiang  Ssu‐hsiang 
Sun Wukong  Sun Wu‐k'ung 
Suyue  Su‐yüeh 
Taiji  T'ai‐chi 
Taijiquan  T'ai‐chi‐ch'üan 
Taijitu  T'ai‐chi‐t'u 
Taipingdao  T'ai‐p'ing Tao 
Taipingjing  T'ai‐p'ing Ching 
Taiqing  T'ai‐ch'ing 
Taishan  T'ai‐shan 
Taishang Daojun  T'ai‐shang Tao‐chün 
Taishang Ganyingpian  T'ai‐shang kan‐ying P'ien 
Taishi  T'ai shih 
Taixi  T'ai‐hsi 
Taixu  T'ai‐hsü 
Taiyi  T'ai‐i 
Taiyidao  T'ai‐i Tao 
Taiyi Jinhua Zongzhi  T'ai‐i Chin‐hua Ts chih 
Taiyue dadi  T'ai‐yüeh ta‐ti 
Tanhuang  Tan‐huang 
Tanjing  T'an‐ching 
Tanluan  T'an‐luan 
Tantian  Tan‐t'ien 
Taoxin  Tao‐hsin 
Taoxuan  Tao‐hsüan 
Tian  T'ien 
Tianchi  T'ien‐chih 
Tian Fang  T'ien Fang 
Tiangu  T'ien‐ku 
Tianming  T'ien‐ming 
Tianshang Shengmu  T'ien‐shang She: mu 
Tianshi  T'ien‐shih 
Tiantai  T'ien‐t'ai 
Tiantiaoshu  T'ien‐t'iao shu 
Tianwang  T'ien wang 
Tiaoqi  T'iao‐ch'i 
Tong Zhongshu  T'ung Chung‐sh 
Tudi  T'u‐ti 
Tutanzhai  T'u‐t'an chai 
Waidan  Wai‐tan 
Waiqi  Wai‐ch'i 
Wang Anshi  Wang An‐shih 
Wangbi  Wang‐pi 
Wang Yangming  Wang Yang‐ming 
Wei  Wei 
Wei Huazun  Wei Hua‐tsun 
Weituo  Wei‐t'o 
Weizheng  Wei Cheng 
Wenchang  Wen‐ch'ang 
Wenshushili  Wen‐shu‐shih‐li 
Wuchang  Wu‐ch'ang 
Wude  Wu‐te 
Wuji  Wu‐chi 
Wujitu  Wu‐chi‐t'u 
Wulun  Wu‐lun 
Wumenguan  Wu‐men‐kuan 
Wumen Huikai  Wu‐men Hui‐k'ai 
Wuqinxi  Wu‐ch'in‐hsi 
Wushan  Wu‐shan 
Wushih Qihou  Wu‐shih Ch'i‐hot 
Wutaishan  Wu‐t'ai‐shan 
Wutoumidao  Wu‐tou‐mi Tao 
Wuwei  Wu‐wei 
Wuxing  Wu‐hsing 
Wuzu Fayan  Wu‐tsu Fa‐yen 
Wuzhenbian  Wu‐chen Pien 
Wuzong  Wu‐tsung 
Hsien  
Xi'an  Sian 
Xi'an Fu  Hsi‐an Fu 
Xiang  Hsiang 
Xiangyan Zhixian  Hsiang‐yen Chih‐hsien 
Xiantian  Hsien‐t'ien 
Xiao  Hsiao 
Xiao Jing  Hsiao Ching 
Xin  Hsin 
Hsing  
Xingqi  Hsing‐ch'i 
Xinxing  Hsin‐hsing 
Xinxinming  Hsin‐hsin‐ming 
Xi Wang Mu  Hsi Wang Mu 
Xiyun  Hsi‐yün 
Xuansha Shibei  Hsüan‐sha Shih‐pei 
Xuantian Shangdi  Hsüan‐t'ien Shang‐ti 
Xuanxue  Hsüan‐Hsüan‐Hsüeh 
Xuanzang  Hsüan‐tsang 
Xuedou Chongxian  Hsüan‐tou Ch'ung‐hsien 
Xuefeng Yicun  Hsüeh‐feng I‐ts'un 
Xu Gaoseng zhuan  Hsü Kao‐seng chuan 
Xun Qing  Hsün Ch'ing 
Xunzi  Hsün Tzu 
Xutang zhiyu  Hsü‐t'ang Chih‐yü 
Yang  Yang 
Yangqi Fanghui  Yang‐ch'i Fang‐hui 
Yangqizong  Yang‐ch'i‐tsung 
Yangshan Huiji  Yang‐shan Hui‐chi 
Yangshen  Yang‐shen 
Yangsheng  Yang‐sheng 
Yangxing  Yang‐hsing 
Yangzhu  Yang Chu 
Yanqi  Yen‐ch'i 
Yanton Chuanhuo  Yen‐t'ou Ch'uan‐huo 
Yayue  Ya‐yüeh 
Yichuan  I‐ch'uan 
Yiguandao  I‐kuan Tao 
Yijing  I‐Ching, Yi Ching 
Yijing  I‐Ching 
Yikong  I‐k'ung 
Yingzhou  Ying‐chou 
Yinxiang  Yin‐Hsiang 
Yinyang  Yin‐yang 
Yinyuan  Yin‐yüan 
Yixuan  I‐Hsuan 
Yongjia Xuanchue  Yung‐chia Hsüan‐chüeh 
Yuanqi  Yüan‐ch'i 
Yuanshi tianzun  Yüan‐shih t'ien‐tsun 
Yuanwu Keqin  Yüan‐wu K'o‐ch'in 
Yuanzhuejing  Yuan‐chueh‐ching 
Yuhuang  Yü‐huang 
Yu Ji  Yü Chi 
Yuangang  Yün‐kang 
Yungan Tansheng  Yün‐yen T'an‐sheng 
Yunji Qiqian  Yün‐chi Ch'i‐ch'ien 
Yunmen Wenyang  Yün‐men Wen‐yen 
Zengzi  Tseng‐tzu 
Zhang Boduan  Chang Po‐tuan 
Zhang Daoling  Chang Tao‐ling 
Zhang Guolao  Chang Kuo‐Iao 
Zhang Jue  Chang Chüeh 
Zhang Ling  Chang Ling 
Zhang Lu  Chang Lu 
Zhangsanfeng  Chang san‐feng 
Zhang Tianshi  Chang T'ien Shih 
Zhang Xian  Chang Hsien 
Zhang Xiu  Chang Hsiu 
Zhaozhou Congshen  Chao‐chou Ts'ung‐shen 
Zhengguan  Cheng‐kuan 
Zhengyi  Cheng‐i 
Zhengyidao  Cheng‐i tao 
Zhenren  Chen jen 
Zhenyan  Chen‐yen 
Zhi  Chih 
Zhidun  Chih‐tun 
Zhi Daolin  Chih Tao‐lin 
Zhiguan  Chih‐kuan 
Zhiyi  Chih‐i 
Zhizhe  Chih‐che 
Zhongguoshi  Chung‐Kuo‐Shih 
Zhongjiao  Tsung‐Chiao 
Zhonglizhuan  Chung‐li Chuan 
Zhongyang  Chung‐yang 
Zhong Yong  Chung Yung 
Zhong Yuan  Chung Yüan 
Zhou  Chou 
Zhou Dunyi  Chou Tun‐(y)i 
Zhou Lianqi  Chou Lien‐ch'i 
Zhu  Chu 
Zhuangzhou  Chuang chou 
Zhuangzi  Chuang‐tzu 
Zhuhong  Chu‐hung 
Zhu Xi  Chu Hsi 
Zi  Tzu 
Zi Si  Tzu Ssu 
Zong  Tsung 
Zongmi  Tsung‐mi 
Zongronglu  Ts'ung‐Jung Lu 
Zou Yan  Tsou Yen 
Zuochan  Tso‐ch'an 
Zuowang  Tso‐wang 
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Chern, ShiingShen
ShiingShen Chern, 1911–2004, ChineseAmerican mathematician, b. Kashing (now Jiaxing), China, D.Sc. Hamburg, 1936. While undertaking graduate studies in China (1932–34), Chern developed what became a lifelong interest in differential geometry. Pioneered in the 19th cent. by Carl Friedrich Gauss in his studies of curves and surfaces, differential geometry received little attention among mathematicians until the 1930s and 40s, but Chern transformed this dormant branch of mathematics into a vibrant area of study. Studying the curvature of surfaces in spaces with more than three dimensions, he devised mathematical quantities that he called characteristic classes—now known as Chern classes—that differentiated different types of surfaces. The fields on which he had the greatest impact, global differential geometry and complex algebraic geometry, are fundamental to many areas of mathematics and theoretical physics, and his work at the foundation of gauge theory and string theory, among the most important developments of modern theoretical physics.
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