**Chuan-Chih Hsiung** describes his ancestors in [2]:-

For many generations my ancestors lived in a small village called Shefong, which is about ten miles from Nanchang, the capital of the province of Jiangsi. The village was located in a picturesque setting, bordered by a shimmering lake in the front and short blue-shaded mountains in the rear.

My ancestors were all farmers until my grandfather's generation when his only brother and he studied the works of Confucius and other literary scholars. Unfortunately they died within two years of each other, both in their mid thirties. At that time, my father [Mu-Han Hsiung], the second of three sons, was only eight years old. Fortuitously, a distant uncle considered my father to be especially intelligent and decided to give him the education that my grandfather had received.

Mu-Han Hsiung received a good education at the Advanced School in Nanchang and there he studied mathematics from English textbooks. After four years of study he graduated and was appointed as vice-principal and mathematics teacher at the new Jiangsi Provincial First High School. Mu-Han Hsiung married Tu Shih and they had four sons and a daughter who was the youngest in the family. Chuan-Chih was the third of the four sons. The second of Mu-Han and Tu Shih's sons, C Y Hsiung, also went on the become a professor of mathematics writing books such as *Elementary theory of numbers* (1992). Chuan-Chih's family moved from Shefong to Nanchang when he was three years old. Chuan-Chih was taught mathematics by his father who gave him a great love of the subject. He graduated from the Jiangsi Provincial First High School in Nanchang and, after sitting a highly competitive university entrance examination, became a mathematics student at the National Chekiang University (founded in 1897) in Hangchow. He graduated in 1936 after being taught by Professor Buchin Su. He began undertaking research in mathematics on topics suggested by Buchin Su. For example he published *On the curvature form and the projective curvatures of curves in space of four dimensions* (1940). In this Hsiung expresses certain invariants of curves in a space of four dimensions found by Buchin Su, and published by him in 1937, in terms of double ratios of covariant points. Hsiung's paper *On the plane sections of the tangent surface of a space curve* (1940) was reviewed by Guy Grove of Michigan State University who wrote:-

This paper concerns itself with the study of the sections of the tangent developable surface of a twisted curve through the tangent line of the curve. The problem was previously studied by Buchin Su[(1933)and(1937)], making use of certain osculants introduced by Bompiani[(1926)]. It is found among other things that among the plane sections of the tangent surface there exists one and only one whose seven-point cuspidal cubic also has eight-point contact; there are only two sections for which the eight-point cubic through the cusp of the seven-point cuspidal cubic has nine-point contact with the plane section; as the sectioning plane revolves about the tangent line the locus of the cusp of the seven-point cuspidal cubic is a twisted cubic; the locus of the line of cusps of the six-point cuspidal cubics is a quadric cone; the locus of the cuspidal tangent is a cubic ruled surface having the tangent to the space curve as a triple line; the locus of the seven-point cuspidal cubic is a surface of order six.

Guy Grove not only reviewed this paper but also Hsiung's papers *Sopra il contatto di due curve piane* (1940), *The canonical lines* (1941), *A graphical construction of the sphere osculating a space curve* (1941), *On the curvature form and the projective curvatures of a space curve* (1942), *Asymptotic ruled surfaces* (1943), *Projective differential geometry of a pair of plane curves* (1943), *Theory of intersection of two plane curves* (1943), *An invariant of intersection of two surfaces* (1943), and *Projective invariants of a pair of surfaces* (1943). Grove was very impressed with Hsiung's mathematics and tried to arrange for him to come to Michigan State University to study for a doctorate. Hsiung had been keen to continue his education in the United States after he graduated but he had been prevented by the war. In 1937 war had broken out between China and Japan. Chekiang province was in the front line for Japanese attacks and many people left the province and moved to safer areas further west. By 1938 the Japanese occupied much of Chekiang province and Hsiung could not travel to the United States.

Although we listed several of Hsiung's papers between 1940 and 1943 there was one which we did not mention since it was rather different from the rest. This was joint work with his colleague Fu Traing Wang at the National University of Chekiang and involved a study of Tangrams, an ancient Chinese puzzle consisting of seven tiles which can be assembled into a variety of geometric and decorative forms. They proved in 1942 that no more than 13 different convex Tangrams can be formed.

In 1943 Grove arranged for Hsiung to become a teaching assistant at Michigan State University so that he would have sufficient financial support to work for his doctorate. However the war between Japan and China still made it impossible for Hsiung to travel to the United States. By this time China was receiving considerable support from the United States in its war effort but Japan still held most of eastern China. Only after Japan surrendered in May 1945 was it possible for China to begin to try to reunite the country but by this time it looked as though a civil war with the Communists could not be avoided. Only in 1946 was Hsiung able to leave China and begin working with Grove at Michigan State University. He was awarded his doctorate in 1948 for his thesis *Rectilinear Congruences.* In fact he had the distinction of becoming the first student to be awarded a Ph.D. by Michigan State University.

After the award of his doctorate, Hsiung was appointed as an instructor at Michigan State University and held this for two years until 1950 after which he spent the winter semester as a visiting lecturer at Northwestern University in Evanston. Hassler Whitney then invited Hsiung to become his research assistant at Harvard and Hsiung took up this position in the spring of 1951. He wrote [2]:-

I benefited very much from my visit to Harvard; I was able to learn the latest developments in mathematics.

Whitney went to the Institute for Advanced Study in the autumn of 1952 and Hsiung was offered an assistant professorship at Lehigh University in Bethlehem, Pennsylvania. The university had been founded in 1865 by Asa Packer, an industrialist and philanthropist, and Hsiung was to spend the rest of his career there. He was promoted to associate professor in 1955 and then full professor in 1960. He retired, receiving the title professor emeritus, in 1984.

We have already indicated the range of Hsiung's early work on projective geometry. After he had spent time in Harvard as Whitney's research assistant, he began studying more global geometry problems. Topics he then investigated include two-dimensional Riemannian manifolds with boundary (uniqueness and isoperimetic inequalities), groups of conformal transformations of a compact Riemannian manifold, curvature and characteristic classes, complex structure, and isospectral almost-*L*-manifolds. His interests are illustrated by looking at the various textbooks he has published. *A first course in differential geometry* published in 1981 was reviewed by O Kowalski who writes that it:-

... is designed as a course of classical differential geometry for beginning graduate students or advanced undergraduate students. Unlike most classical books on the subject, more attention is paid to the relationship between local and global properties. In fact, the title of the book might well be A first course in differential and integral geometry.

Two further texts, written after he retired, are *Almost complex and complex structures* (1995) and *A first course in differential geometry* (1997). The first of these is a monograph which presents many of Hsiung's own results, in particular including his important work on the nonexistence of a complex structure on *S*^{6}. The second of these books has been reviewed by Man Chun Leung who writes:-

The book is designed to introduce differential geometry via the study of curves and surfaces inE^{3}.It begins with a review on point-set topology, multi-dimensional calculus and linear algebra. With the fundamental material in place, the author discusses the theory of smooth curves, including the Frenet formulas, the isoperimetric inequality and the four-vertex theorem. Local theory of surfaces is presented, with the emphasis on fundamental forms, curvatures, ruled and minimal surfaces. The global theory of surfaces inE^{3}is studied in the last chapter. The central themes are the Gauss-Bonnet formula and uniqueness theorems for the Minkowski and Christoffel problems. The author presents the discussion in a precise, direct and step-by-step manner. Most sections end with a carefully selected set of exercises, some of which supplement the content of the section. The book provides a solid introduction to differential geometry of curves and surfaces. It also includes many short and valuable remarks on the classical literature of the subject.

There is one very important contribution which Hsiung has made which we have not mentioned yet. This is his founding of the Journal of Differential Geometry in 1967. He acted as editor-in-chief of the journal from the time of its foundation. Hsiung writes [2]:-

Under the influence of this journal, differential geometry has become a very active branch of mathematics, with scope far exceeding its former classical one.

Man Chun Leung writes:-

It is not necessary to heap accolades upon this contribution - the journal has long been recognized as a forum of publication of the finest papers in differential geometry.

**Article by:** *J J O'Connor* and *E F Robertson*